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Optimal Design of a Coreless Axial Flux
Permanent Magnet Synchronous Generator for the
Wind Power Generation
By
Junaid Ikram
CIIT/SP11-PEE-001/ISB
PhD Thesis
In
Electrical Engineering
COMSATS Institute of Information Technology
Islamabad-Pakistan
Spring, 2017
ii
COMSATS Institute of Information Technology
Optimal Design of a Coreless Axial Flux
Permanent Magnet Synchronous Generator for the
Wind Power Generation
A Thesis Presented to
COMSATS Institute of Information Technology, Islamabad
In partial fulfillment
Of the requirement for the degree of
PhD Electrical Engineering
By
Junaid Ikram
CIIT/SP11-PEE-001/ISB
Spring, 2017
iii
Optimal Design of a Coreless Axial Flux
Permanent Magnet Synchronous Generator
for the Wind Power Generation
A Post Graduate Thesis submitted to the Department of Electrical Engineering
as partial fulfillment of the requirement for the award of Degree of PhD in
Electrical Engineering.
Name
Registration Number
Junaid Ikram
CIIT/SP11-PEE-001/ISB
Supervisor
Prof. Dr. Nasrullah Khan
Department of Electrical Engineering
COMSATS Institute of Information Technology (CIIT)
Islamabad
viii
DEDICATION
To
This thesis is dedicated to the families of missing and dead
people during the war on terror, my parents and wife,
and
to Prof. Dr. Nasrullah Khan, whose constant support and
encouragement made this research work possible.
ix
ACKOWLEDGEMENTS
The research work and contributions presented in this thesis are accomplished by
the grace of Almighty Allah. The success behind this accomplishment is due to
academic support by my advisor Prof. Dr. Nasrullah Khan. He has always been very
kind and helpful and devoted his precious time whenever I needed some help. His
valuable guidelines, critiques, patience, and support have enabled me to achieve this
prestigious milestone in my life.
I am also grateful to Prof. Byung il Kwon for his kind guidance and support. His
valuable guidelines, comments, suggestions and corrections paved way for my
success. I would like to thank my colleagues for their guidance, help and support
during my stay in South Korea. This is an important milestone in the long journey of
life that will continue.
The prayers of my parents, relatives, and friends were a prime source of
motivation to me during my studies which I wholeheartedly appreciate. The research
work presented herein is conducted at the Department of Electrical Engineering,
COMSATS Institute of Information Technology (CIIT), Islamabad Campus.
Moral, administrative and technical support of Prof. Dr. Shahid A. Khan, Prof. Dr.
M. Junaid Mughal, Dr. Qadeer ul Hassan, Dr. Fasih Uddin Butt, and Dr. Ali Arshad,
have always been available to me during the entire period of this research work. In
addition, I am also thankful to Mr. Muhammad Naeem and Kashif Nazir at the
Graduate Office who have helped me in the official matters whenever required.
Last but not least, I acknowledge international research support initiative program
(IRSIP) and the Hanyang University, South Korea.
Junaid Ikram
CIIT/SP11-PEE-001/ISB
x
ABSTRACT
Optimal Design of a Coreless Axial Flux Permanent Magnet
Synchronous Generator for the Wind Power Generation
Renewable power generation from wind and solar are gaining popularity to
overcome energy crisis nowadays. A lot of advancement has been focused on wind
power generation instead of fossil fuels that are degrading to the environment since
last two decades in order to increase electricity generation, efficiency improvement,
reliability and cost reduction. The generator used in windmill can be an induction
generator (IG), synchronous generator (SG), doubly fed induction generator (DFIG),
radial flux permanent magnet synchronous generator (RFPMSG) and axial flux
permanent magnet synchronous generator (AFPMSG). Furthermore, due to the
variable speed of wind turbine, a fully rated power converter handles the extracted
energy in direct drive systems or a coupled geared system. However, with geared
system, the cost of the overall system increased a lot and proved to be rather less
reliable. In this regard, AFPMSG are most suitable for the direct drive applications
due to its disc shape structure.
The design of AFPMSG is derived from the design of RFPMSG. By using the
desired value of parameters like power, speed, efficiency, number of phases,
frequency, rated voltage and by taking some assumptions, inner and outer diameter of
the rotors can be computed using sizing equation. Furthermore, in order to get balance
three phase output and suitable winding factor a proper combination of the coils and
poles is required. A 1 kW dual rotor single coreless stator AFPMSG, with
concentrated winding is designed by using sizing equation in this research work.
In order to analyze the characteristics of an electric machine analytical method
formed on the solution of Maxwell equations and Finite Element Method (FEM) are
used. The FEM results are more reliable as compared to the analytical method.
However, FEM take long computation time as compared with the analytical method.
This thesis presents a 2D analytical method to calculate the no load voltage of the
coreless dual rotor AFPMSG. Furthermore, to decrease the no load voltage total
harmonic distortion (VTHD), initial model of the coreless AFPMSG is optimized by
using the developed analytical method. The back EMF obtained by using the 2-D
analytical method is confirmed by time stepped 3-D FEM for both the initial and
xi
optimized models. Finally, VTHD, torque ripple and output torque are compared for
the initial and optimized models by using the 3-D FEM. It is demonstrated that the
VTHD and torque ripples of the optimized model are reduced as compared to the
initial model. Optimization by utilizing the 2-D analytical method reduces the
optimization time to less than a minute.
Furthermore, an AFPMSG model to reduce torque ripple is presented in this thesis.
The proposed model uses arc-shaped trapezoidal PMs. The proposed model reduced
cogging torque and torque ripple at the expense of lower average torque. Time
stepped 3-D FEM is performed and the results are compared with the conventional
model. It is demonstrated that the torque ripple of the proposed model is reduced as
compared with the conventional model.
To further improve the performance of the designed machine with proposed
magnet shape, it's PM shape is optimized. The Latin Hyper Cube Sampling (LHS),
Kriging Method and Genetic Algorithm (GA) are introduced and employed in the
proposed machine for the optimization. Asymmetric magnet overhang, interpolar
separation of PMs and axial height of PMs are considered as the design variable for
the optimization. The volume of the PMs is kept equal to the conventional shape
magnet volume during optimization. It is demonstrated that the torque ripple of the
optimized model is reduced and the average torque is increased as compared with the
conventional and proposed models. The optimized model shows improvement in
terms of the quality of the torque along with average output torque.
The proposed coreless AFPMSG presents a suitable alternative to meet increasing
energy demand as compared to the conventional AFPMSG due to its reduced cogging
torque and torque ripple and increased output power and torque. The research work
presented in this thesis seems to be an attractive option in the field of axial flux
machine to be utilized for wind power applications.
xii
TABLE OF CONTENTS
Chapter 1 Introduction ..................................................................................... 1
1.1 Research Background ............................................................................................................ 2
1.2 Significance of the thesis ..................................................................................................... 13
1.3 Main contents of the thesis .................................................................................................. 14
1.4 Thesis organization ............................................................................................................. 14
Chapter 2 Axial Flux Permanent Magnet Machines ................................... 16
2.1 Axial Flux Permanent Magnet Machines Topologies ......................................................... 17
2.2 Winding configuration in AFPM machines ........................................................................ 23
2.3 Summary ............................................................................................................................. 26
Chapter 3 Design and Analysis Procedure.................................................... 27
3.1 Magnet Operating Point ...................................................................................................... 28
3.2 Design Method .................................................................................................................... 30
3.3 Analysis Method ................................................................................................................. 38
3.4 Optimization Method .......................................................................................................... 41
3.4.1 Kriging Method ................................................................................................................... 42
3.4.2 Genetic Algorithm ............................................................................................................... 44
3.5 Summary ............................................................................................................................. 45
Chapter 4 Analysis of AFPMSG with 2-D Analytical Method ................... 46
4.1 Introduction ......................................................................................................................... 47
4.2 2-D Analytical Modeling for Coreless AFPMSG Analysis ................................................ 48
4.2.1 Initial Model ........................................................................................................................ 48
4.2.2 Assumptions ........................................................................................................................ 48
4.2.3 Magnetization of the PMs ................................................................................................... 50
4.2.4 2-D Analytical Method ........................................................................................................ 52
4.2.5 Characteristics Analysis ...................................................................................................... 57
4.3 Optimization of the AFPMSG using 2-D Analytical Method ............................................. 60
4.4 Summary ............................................................................................................................. 64
Chapter 5 Reduction of Torque Ripple in an AFPMSG using Arc Shaped
Trapezoidal Magnets in an Asymmetric Overhang Configuration ............ 65
5.1 Introduction ......................................................................................................................... 66
5.2 Comparison between the proposed and conventional Model .............................................. 66
5.2.1 Proposed Magnet Shape ...................................................................................................... 67
5.2.2 Design Process .................................................................................................................... 68
5.2.3 AFPMSG Conventional and Proposed Models Performance Comparison ......................... 70
5.3 Proposed Model Optimization ............................................................................................. 73
xiii
5.3.1 Selection of Design Variables ............................................................................................. 74
5.3.2 Optimization Process ........................................................................................................... 75
5.3.3 Optimal Design Results ....................................................................................................... 76
5.4 Summary ............................................................................................................................. 80
Chapter 6 Conclusion and Future Work ...................................................... 81
References ..................................................................................................... 84
xiv
LIST OF FIGURES
Figure 1.1 Wind energy system core components ........................................................ 2
Figure 1.2 Michael Faraday's acyclic machine ............................................................. 3
Figure 1.3 Grid connected squirrel cage induction generator ....................................... 4
Figure 1.4 Improved version of grid connected squirrel cage induction generator ...... 4
Figure 1.5 Grid connected wound rotor induction generator ........................................ 5
Figure 1.6 The grid connected DFIG for wind power generation ................................ 6
Figure 1.7 Magnetic materials growth for the energy density (BH)max ...................... 10
Figure 1.8 Classification of the machines according to the direction of the flow
of flux (a) RF machine (b) AF machine (c) TF machine ............................................. 11
Figure 1.9 Inner Rotor PM machine possibilities (a) Surface mounted (b) Surface
inset (c) Interior radial (d) Interior circumferential ..................................................... 12
Figure 1.10 Outer Rotor PM Machine variants (a) with single bond PM (b) with
p pole pairs ................................................................................................................... 12
Figure 2.1 AFPM Machine Topologies (a) SSSR (b) SSDR (c) DRSS (d) MSMR ... 18
Figure 2.2 Axial Flux Machine Topologies Flow Chart ............................................. 19
Figure 2.3 SSSR Axial Flux Machine Topologies (a) Slotted type (b) Slot-less
type ............................................................................................................................... 19
Figure 2.4 DSSR Axial Flux Machine Topologies (a) Slot-less type (b) Slotted
type ............................................................................................................................... 20
Figure 2.5 DRSS AFPM machine Topologies (a) Coreless type (b) Slotted type
(c) Slot-less type .......................................................................................................... 21
Figure 2.6 Flux path in DRSS topologies (a) NN Slot-less Core type (b) NS
Slotted core type (c) NS coreless type ......................................................................... 22
Figure 2.7 Magnet shapes in DRSS AFPM machines: (a) Trapezoidal (b) circular
(c) semi-circular ........................................................................................................... 22
Figure 2.8 Multistack or multistage AFPM machine .................................................. 23
Figure 2.9 Typical winding configurations (a) Overlapping (distributed) (b)
Overlapping (concentrated). (c) Nonoverlapping, all teeth wounds (d)
Nonoverlapping, alternate teeth wound. ...................................................................... 24
xv
Figure 2.10 Winding types in core type structure of AFPM machines (a) Tooth
Wound (ring type) (b) Core Wound (drum type) (c) Ring Type Concentrated (d)
Drum Type Concentrated (e) Drum Type Distributed ................................................. 25
Figure 2.11 Winding types in coreless structure of AFPM machines (a) Single
Layer Concentrated (b) Double Layer Concentrated (c) Triple Layer
Concentrated (d) Triple Layer Wave Winding ............................................................ 25
Figure 3.1 Operating Point of the Magnet .................................................................. 29
Figure 3.2 Basic Magnetic Circuit .............................................................................. 29
Figure 3.3 Flow chart of the design process. .............................................................. 37
Figure 3.4 Optimal points: (a) The boundless domain and function (local minima
and maxima), (b) The bounded domain and function (global minima and
maxima) ....................................................................................................................... 42
Figure 3.5 The flow chart of GA................................................................................. 45
Figure 4.1 Exploded view of the 3-D FEA model of the AFPMSG. .......................... 49
Figure 4.2 Magnetization produced by the PMs. ........................................................ 50
Figure 4.3 Linear representation of the AFPMSG for the lower rotor. ...................... 52
Figure 4.4 Linear representation of the AFPMSG for the upper rotor. ...................... 55
Figure 4.5 Linear representation of the AFPMSG coil region by current sheet. ........ 56
Figure 4.6 (a) Magnetic field's axial component of air region (b) Magnetic field's
circumferential component of air region. ..................................................................... 57
Figure 4.7 (a) Magnetic field's axial component of magnet regions. (b) Magnetic
field's circumferential component of magnet regions. ................................................. 58
Figure 4.8 Armature reaction field. ............................................................................. 59
Figure 4.9 Resultant magnetic field. ........................................................................... 59
Figure 4.10 Back EMF waveforms comparison using 2-D analytical method and
3-D FEA of the initial model. ...................................................................................... 59
Figure 4.11 VTHD trend. ............................................................................................ 60
Figure 4.12 Selected design variables and their optimal values. ................................ 61
Figure 4.13 Optimal design process. ........................................................................... 61
Figure 4.14 Optimized model back EMF comparison with 2-D analytical method
and 3-D FEA. ............................................................................................................... 62
Figure 4.15 Belt Harmonics comparison. ................................................................... 62
Figure 4.16 Flux density distribution plots by 3D-FEA: (a) Initial model (b)
Optimized model. ......................................................................................................... 63
xvi
Figure 4.17 Torque comparison of the initial and optimized model by 3D- FEA. ..... 64
Figure 5.1 PM shapes: (a) Conventional magnet (b) Proposed magnet...................... 67
Figure 5.2 Parameters of the PM shapes: (a) trapezoidal (b) arc-shaped
trapezoidal .................................................................................................................... 68
Figure 5.3 Flow chart of the design process. .............................................................. 69
Figure 5.4 Exploded AFPMSGs with concentrated windings: (a) conventional
model (b) proposed model. .......................................................................................... 70
Figure 5.5 Flux density distribution: (a) entire conventional model (b) coil
region. .......................................................................................................................... 71
Figure 5.6 Flux density distribution: (a) entire proposed model (b) coil region. ........ 71
Figure 5.7 Back EMF waveforms for the conventional and proposed models. .......... 71
Figure 5.8 Cogging torque comparison of the conventional and proposed models. ... 72
Figure 5.9 Torque comparison of conventional and proposed models. ...................... 73
Figure 5.10 Design variables: (a) Asymmetric PM overhang, (b) Top view (c)
Cross-sectional view. ................................................................................................... 74
Figure 5.11 Arc-shaped trapezoidal PM parameters. .................................................. 75
Figure 5.12 Optimal design process. ........................................................................... 76
Figure 5.13 Optimized model flux density distribution. ............................................. 76
Figure 5.14 Optimized model back EMF. ................................................................... 77
Figure 5.15 Cogging torque comparison of the proposed and optimized models. ..... 77
Figure 5.16 Torque comparison of the proposed and optimized models. ................... 78
Figure 5.17 Output power comparison of the proposed and optimized models. ........ 78
xvii
LIST OF TABLES
Table 1.1 A comparison of various induction generator types. .................................... 7
Table 1.2 Magnet performance comparisons .............................................................. 10
Table 3.1 Standard values for TRV ............................................................................. 36
Table 3.2 Selection of electrical loading and current density ..................................... 37
Table 4.1 The AFPMSG parameters ........................................................................... 49
Table 4.2 Initial model performance comparison using 2-D analytical and 3-D
FEA .............................................................................................................................. 60
Table 4.3 Optimized model performance comparison with 2-D analytical method
and 3-D FEA ................................................................................................................ 63
Table 4.4 Initial and optimized model comparison ..................................................... 64
Table 5.1 Conventional and proposed models parameters .......................................... 68
Table 5.2 Performance comparison between the conventional and proposed
models of coreless AFPMSGs ..................................................................................... 73
Table 5.3 Comparison of design parameter ................................................................ 79
Table 5.4 Comparison of performance parameters ..................................................... 79
xviii
LIST OF ABBREVIATIONS
τp Pole pitch
B Flux density
Br Remanent flux density
d Interpolar separation
f Frequency
Axial length of the magnet
Inner radius of the rotor
Outer radius of the rotor
H Magnetic field intensity
i Instantaneous current in armature conductor
M Magnetization
Order of the harmonics
Turns per phase
p Number of pole pairs
P Number of poles
L Axial height of the machine
m Number of phases
µo Permeability of free space
µr Relative permeability of the material
µ Permeability of the material
αp Pole arc to pole pitch ratio
Φ Magnetic scalar potential
R Mean radius of the rotor
i Unit vector along x-axis
j Unit vector along y-axis
Xc Width of a phase band of armature winding
αc Width of the coil of armature winding
y axial height
x circumferential distance
Hx circumferential components of the magnetic field intensity
Hy axial component of the magnetic field intensity
xix
By axial component of the magnetic flux density
Bx circumferential component of the flux density
Eb back EMF
VTHD Voltage total harmonic distortion
Hm magnetic field intensity of the magnet
Hg magnetic field intensity of the air region
lg length of the air gap region
Bg air gap flux density
Bm operating flux density of the magnet
Am area of the magnet
Ag area of the air gap region
PC permeance coefficient
µrm relative permeability of the magnet
Ω speed of the machine in revolution per minute
ns speed of the machine in revolution per second
ωr speed of the machine in radian per second
ωe electrical speed in radian per second
S number of the stator coils
Cn coils per phase
Q number of coils per poles
q1 number of coils per poles per phase
Nc number of turns per coils
nc number of conductors per coils
aw number of parallel conductors
τci inner coil pitch
τco outer coil pitch
τca mean coil pitch
τpi inner pole pitch
τpo outer pole pitch
τpa mean pole pitch
Di inner diameter of the rotor disc
Do outer diameter of the rotor disc
Da mean diameter of the rotor disc
kd ratio between rotor inner diameter to the rotor outer diameter
xx
Tm radial length or thickness of the magnet
Ri inner radius of the rotor disc
Ro outer radius of the rotor disc
Ra mean radius of the rotor disc
Ly length of the rotor disc or back iron
Bmax maximum allowable flux density of the rotor back iron
Lw coil length in axial direction
kw winding factor
kp pitch factor
kd1 distribution factor
β ratio between coil pitch and pole pitch
Am electrical loading
α ratio between averages and peak magnetic flux densities
φp flux per pole
Ψ flux linkage
Pout output power
τout output power
η efficiency
cosφ power factor
TRV torque per unit rotor volume
σ shear stress
U magnetic scalar potential
A magnetic vector potential
PM permanent magnet
AFPM axial flux permanent magnet
RFPM radial flux permanent magnet
TFPM transverse flux permanent magnet
DFIG double fed induction generator
PMSM permanent magnet synchronous machine
BLDC brushless DC
BDFIG brushless doubly fed induction generator
SCIG squirrel cage induction generator
FSWT fixed speed wind turbine
VSWT variable speed wind turbine
xxi
WRIG wound rotor induction generator
PCC point of common coupling
GSC grid side converter
RSC rotor side converter
VRM variable reluctance machine
SRG switch reluctance generator
BDFRG brushless doubly fed reluctance generator
PMBLAC permanent magnet brushless AC
PMBLDC permanent magnet brushless DC
FEM finite element method
FEA finite element analysis
SSSR single stator single rotor
DSSR double stator single rotor
SSDR single stator double rotor
MSMR multi stator multi rotor
AFIR axial flux inner rotor
THD total harmonic distortion
xxii
LIST OF PUBLICATIONS AND PRESENTATIONS
Journal Publications
1. Junaid Ikram, Nasrullah Khan, Qudsia Junaid, Salman Khaliq, Byung il Kwon,
“Analysis and Optimization of the Axial Flux Permanent Magnet Synchronous
Generator using an Analytical Method”, Journal of Magnetics, (Article ID:
E2017-21). Accepted
2. Junaid Ikram, Nasrullah Khan, Salman Khaliq and Byung il Kwon, “Reduction
of Torque Ripple in an Axial Flux Generator Using Arc Shaped Trapezoidal
Magnets in an Asymmetric Overhang Configuration”, Journal of Magnetics, 21
(4), pp. 577-585, Dec 2016.
3. Junaid Ikram, Nasrullah Khan, Byung il Kwon, “Improved Model of the Iron
Loss for the Permanent Magnet Synchronous motors”, Journal of international
conference on electric machine and system, 1(2), pp. 10-17, 2012.
Conference Publications
1. Junaid Ikram, Qudsia Junaid, Byung il Kwon, “Improved Model of the Iron
Loss for the Permanent Magnet Synchronous Motors”, ICEMS, Incheon Korea,
October 10-13, 2010.
2. Qudsia Junaid, Junaid Ikram, You Yong-min and Byung-il Kwon, “Analytical
Analysis and Optimization of the Double Sided AFPMSG”, CEFC, 2010,
Chicago, USA, May 11-13, 2010.
Symposium Presentation
1. Junaid Ikram, and Nasrullah Khan, “Design and analysis of axial flux permanent
magnet synchronous generator for wind power generation”, In: Symposium on
Research Innovation in IT & Engineering (RIITE), April 2013, COMSATS
Institute of Information Technology, Attock, Pakistan.
1
Chapter 1 Introduction
2
1.1 Research Background
The utilization of electric machine has increased in many applications during the
past few decades. The most common applications of these include electric vehicles,
home appliances, audio and video devices, computers, fuel pumps, power generation,
aircrafts and industrial drives. The depletion of fossil fuels, environmental concerns
and the energy crisis has motivated the researchers to find economical and
environment friendly solutions for generating electrical energy. Recently, renewable
power generation from the wind's kinetic energy has gained popularity because of its
environment friendly nature [1]. The generators that are used most commonly in
windmills include squirrel cage induction generator (SCIG), electrically excited
synchronous generator (EESG), doubly fed induction generator (DFIG) and
permanent magnet synchronous generator (PMSG) [2-5]. Squirrel cage induction
machines remained the most popular electrical machines due to its robust structure,
low cost and moderate reliability, during the 20th century. However, their
disadvantages are low efficiency, the need for an AC excitation source and low power
factor [6-8]. The PM synchronous machines are widely used in past few decades due
to their brushless operation, compact structure and high power density [9, 10].
Wind turbine produces mechanical power by altering the kinetic energy of the
wind. The electrical power is produced by the generator from this mechanical power.
A Gearbox is used for matching the turbine speed with the generator rated speed. The
power electronic convertor converts the generator voltage into DC and then into AC
to connect this with the grid. The classification of the wind turbines is according to
the rotational speed, axis of rotation and drive train. There are two main types of wind
turbines, according to the speed; fixed speed drive and variable speed drive. The wind
turbines are categorized into horizontal axis and vertical axis according to the
rotational axis. The direct drive and geared drive are the main types of wind turbine
according to the drive train classification. The main components of the wind energy
system consist of wind turbine, electric machine and convertors, as shown in the
Figure 1.1 [11-14].
Figure 1.1 Wind energy system core components
Wind Wind TurbineGear box (optional) Generator Power Converter
(optional)
Transformer
3
Michael Faraday built acyclic machine in 1831, as shown in Figure 1.2 [15]. It is
the world's first electric machine [16]. It is also known as homo-polar machine
because the polarity of the magnetic field does not change as compared with
conventional DC machine. The first patent in electric machine was claimed by
Davenport published in 1837, entitled the improvement in electromagnetic machines.
Figure 1.2 Michael Faraday's acyclic machine
There are two types of electric machines regarding the type of field excitation
system, i.e. wound field and PM-field electric machines. Generally, in wound field
machines, the electromagnets provide rotor field excitation, whereas in PM-field
machines the permanent magnets provide rotor field excitation. The most prominent
types of wound field machine are synchronous, DC and induction machines.
Squirrel cage induction machines remained the most popular electrical machines
due to its robust structure, low cost and moderate reliability, during the 20th century.
However, their disadvantages are low efficiency, the need for an AC excitation source
and low power factor as compared to the DC and synchronous machines [6-8]. As the
name indicates, the rotor design of the SCIG has squirrel cage like structure, where
the solid conducting bars are used as winding. These conducting bars are made from
either copper or aluminum, which are shorted from both sides via end rings. Mostly,
these bars are slightly skewed (one slot pitch) in structure to reduce cogging torque
and hence torque ripple and noise [17]. The SCIG machine is basically a type of
fixed speed induction machine that is used for power generation from wind. For fixed
wind power generation applications, SCIG are used most commonly.
For wind power generation, the fixed speed SCIG based wind energy conversion
system mainly consists of the SCIG, reactive power compensation capacitors and the
soft starter. The stator is directly connected to the grid , whereas, the rotor is coupled
+ Brush
Rotating Copper Disc
- Brush
4
with three stage gear box mechanically (Danish concept) which enables stall regulated
wind turbines to run at fixed speed. When power is supplied to the stator from the
grid, rotating magnetic field develops across the air gap [18]. Similarly, the rotor gets
energized when there develops a negative slip, i.e. the rotor moves at a higher speed
(super-synchronous) than the synchronous speed.
One of the disadvantages of the SCIG is that it consumes reactive power from the
utility grid consistently because of its magnetizing reactance, and this leads to low full
load power factor which is an undesirable situation particularly for weak grids [18].
Therefore, SCIG uses capacitor bank for its reactive power compensation, as shown in
Figure 1.3 [19]. The soft starter here used is for smoothing of the inrush current. The
SCIG operates at constant (in fact narrow range) of wind speeds, whereas the wind
involves wide range of speeds, so the maximum power output from SCIG is not
expected and this ultimately results in overall low power efficiency.
Figure 1.3 Grid connected squirrel cage induction generator
There are some advanced versions of the SCIG that are used in wind energy
conversion system, such as SCIG with two winding sets, where first winding is used
for fixed wind speeds, whereas the other winding set operates at variable wind speed,
hence this way the efficiency of the SCIG based wind energy conversion system
(WECS) is improved. Some other SCIG designs use back to back power converters as
an alternate for capacitor bank [20]. By using this converter technology, SCIG is now
able to harness more energy as compared to its conventional design, but the cost of
this power converter is higher than the conventional capacitors. The SCIG topology
based on power converter is shown in Figure 1.4 [21].
Figure 1.4 Improved version of grid connected squirrel cage induction generator
Wind Turbine Gear Box SCIG Soft Starter Transformer
Compensating
Capacitor
Grid
Wind Turbine Gear Box SCIG AC/DC Transformer GridDC/AC
5
Since the power efficiency of fixed speed WECS is relatively low therefore,
limited variable wind speeds WECS with higher power efficiency are introduced. The
wound rotor induction generator (WRIG) is a type of induction machine which
operates at limited variable wind speeds for wind power generation. The stator
configuration of the WRIG is similar to the SCIG design, however, the rotor windings
are brought out via slip rings and brushes. The rotor component is punched by stacked
laminations and fitted directly onto the shaft. For wind energy applications, the stator
is coupled to the electrical grid, whereas, the rotor has a variable resistance which is
commonly known as Optislip. By changing the rotor resistance, the variable speed
operation (slip) of the WRIG is regulated and this way the output power of the
generator is controlled. Since WRIG needs reactive power for generator excitation, so
a capacitor bank for reactive power compensation is added to the circuit [22] as
shown by the Figure 1.5 [19]. Generally, the typical speed range for WRIG is 0-10%
above synchronous speed [23].
Figure 1.5 Grid connected wound rotor induction generator
In recent times, variable speed wind turbines (VSWT) are more prominent and
have received more attention because of their advantages over fixed speed wind
turbines (FSWT). Since wind is non-linear and non-stationary in nature, so the power
output from FSWT has fluctuations and variations in it which ultimately results in
poor power quality upon grid integration. This needed a system which could
incorporate the drawbacks associated with FSWTs, and hence the variable speed wind
turbines were introduced, which mitigated the issues that were found in FSWTs.
Doubly Fed Induction Generator (DFIG) is one such VSWT which ensures maximum
power capture and made it possible to transfer that maximum power to the electrical
grid with varying wind speeds. Therefore, DFIGs are now generally used for
manufacturing large scale turbines.
Wind Turbine Gear Box WRIG Soft Starter Transformer
Compensating
Capacitor
Grid
Variable Resistor
6
For wind power system, the stator in DFIG is connected to the utility grid through
transformers, while the windings in rotor are connected to the slip rings and carbon
brushes that are mainly used to transfer the power to or from the grid through bi-
directional back to back voltage source converters. The converters regulate the rotor
current, frequency and phase angle shifts. In other words, by using these converters,
the slip power is controlled in general. The converters consist of Rotor Side Converter
(RSC), Grid Side Converter (GSC) and a DC link. The RSC controls the torque or
speed and power factor of the DFIG, whereas the GSC is used to minimize the voltage
ripples caused by DC link capacitor. The slip range for DFIG is +30% above the
synchronous speed that results in maximum power extraction, reduced mechanical
stress and power fluctuations and better reactive power control [24].
For DFIG in sub-synchronous mode, the RSC acts as an inverter and the GSC acts
as a rectifier, and in such scenario the active power from the grid enters into the rotor.
Reverse is true for converters operation in super-synchronous condition, the only
difference is that, here the power flows from both stator and rotor to the electrical
network (a grid) [18]. The schematic diagram for the grid connected DFIG is shown
in Figure 1.6 [25].
Figure 1.6 The grid connected DFIG for wind power generation
In conventional DFIG, use of brushes and slip rings requires consistent
maintenance which is undesirable situation from machine’s operational point of view.
Therefore, brushless designs are more preferred in recent times because of their low
maintenance activity and robustness. The Brushless doubly fed induction generator
(BDFIG) is a variational design for typical DFIG which is more preferable for
offshore wind applications [26]. The BDFIG can be constructed either 1) using a
single stator with double windings where both stator layers have different number of
poles; or 2) by using two cascaded induction machines; however, the working
Wind Turbine Gear Box DFIG AC/DC Transformer GridDC/AC
SCR
Rcrow
7
principle is same for both configurations. Let’s consider the BDFIG that consists of
two cascaded IMs (wound rotor) which includes the main power machine and the
control machine. The main power IM is coupled to the grid directly, whereas the
control IM is connected to the grid via back to back electronic power converters. The
rotor circuit of the two IMs is connected in such a way that their combine torque
effects are added to enhance the generator over all torque rating and this way
operational range of the system is enhanced [27]. The BDFIG operates at wide range
of wind speeds and thus higher energy yield, which makes it appealing for large scale
wind turbines specifically for offshore wind turbines, but the control mechanism for
BDFIG is relatively complex as compared to typical DFIG [28]. The merits and
demerits of the three induction machines are summarized in the Table 1.1.
Table 1.1 A comparison of various induction generator types.
Generator
Type
Advantages Disadvantages
Fixed speed
IM ( SCIG)
Simple mechanical
design
Robust and Rugged
Low maintenance
Cheap
Low energy yield
Output power fluctuations
External reactive power
compensator is required
High mechanical stress
High gear losses
Limited
speed IM
( WRIG)
Operates in limited
speed variation
Slip rings and brushes
may replace optical
coupling
Speed variation is dependent
on variable rotor resistance
Reactive power compensator
is needed
Variable
speed IM
(DFIG)
Maximum power
extraction
Wide range of speed
variations
No external power
compensator is
required
Generally used for
large scale wind
turbines
Complicated control
mechanism
Complicated converter
design
Multistage gear box and slip
rings needed
Overall cost of the system is
high
Variable reluctance machines (VRMs) are the synchronous machines that are used
with variable speed WECS. These machines are more robust and simpler than PM
machines since they do not have permanent magnets in them. Switch reluctance
generator (SRG) is the most common type of VRMs which has attracted researchers
8
attention to explore the potentials of the switch reluctance machine. In SRG, the stator
has the phase windings while the rotor consists of steel laminations instead of
conventional windings or permanent magnets, this makes SRG suitable for high
temperature environment and high speed applications. Moreover, absence of windings
and magnets results in low inertia, which enables SRG to respond rapidly at speed and
load variations [29]. The torque characteristics of switch reluctance machine (SRM)
depends upon variations in inductance or reluctance produced by the interaction of the
stator and rotor poles. During fully alignment of the stator and rotor poles, the
inductance is maximum and it goes minimum when stator and rotor are fully
misaligned. The excitation for SRG is made when the rotor is passing by the stator
(during the time when the reluctance is decreasing), whereas for motoring, excitation
is made when reluctance is increasing. SRM are good alternative to synchronous and
induction machines, however these machines suffer from current commutation and
huge turn off inductance due to lack of separate field excitation source, this ultimately
reduces the overall torque characteristics of these machines. To overcome this issue,
auxiliary compensation windings are introduced in the stator or rotor. There are
various other types of VRM depending upon their winding configurations on stator or
rotor, however VRM with auxiliary winding is more robust and stable mechanically
[29].
An alternative design for BDFIG with improved performance is the brushless
doubly fed reluctance generator (BDFRG) which is a one type of variable reluctance
machine. As the name implies, BDFRG does not contain magnets, brushes and rotor
circuits, these features enhances its robustness, controllability and low maintenance,
which is the ultimate aim of any WECS for maximum power extraction. Unlike
brushless DFIG, the BDFRG uses reluctance rotor in place of wound rotor and is
more efficient than BDFIG in many aspects, such as, it offers higher efficiency,
reliability and easier control system as compared to the BDFIG. The BDFRG stator
consists of two stator windings with different number of poles; whereas, the rotor
poles are defined by the number of stator poles (half of the sum of both primary and
secondary windings pole pairs). The torque is produced by the relative
motion/position of the rotor with respect to stator pole winding which cause
inductance or reluctance variations and this consequently results in torque production
[30]. The two stator windings which are known as primary/main power winding and
9
secondary/control winding. The primary winding is directly connected to the grid,
whereas the control winding is connected to the grid via power electronic converter.
One of the prominent features of the BDFRG is that it can also work as typical
synchronous and asynchronous machines. In former case, the secondary windings of
the stator are connected with a DC source while the stator secondary windings in later
are shorted that serves as fail safe mode for situations like inverter failure. This mode
may also be used as starting the system [30]. Variants of BDFRMs such as RF-
BDFRM and AF-BDFRM are now getting more attention because of their high
performance specifically the AF-BDFRM which is applicable in locations where
higher torque density is needed.
The ability to supply reactive power and eradication of slip power loss are the main
advantages of synchronous machines while comparing with the induction machines.
Furthermore, the synchronous machines are also favored due to the reduced weight,
inertia and volume as compared to the DC machine for the same power rating.
Furthermore, for the generation of electrical power, the synchronous machines are
used most commonly in the industry due to its ability to supply reactive power and its
operation close to the unity power factor. However, disadvantages of the machine
with electrical excitation include increased volume, increased copper losses and need
of carbon brushes while comparing with PM-field machine. In addition, brushless
configurations are preferred due to its low cost and robustness [31].
The introduction of PMs to replace electromagnetic poles results in compact
synchronous machines. Recently the popularity of PM machines are becoming more
and more due to the decrease in the cost of the permanent magnet. The PM machines
have high efficiency due to the elimination of field excitation copper losses. These
machines also have higher torque density and smaller volume. Other benefits of the
PM machines include brushless operation, and high power density. The most
prominent type of PM excited machine is a permanent magnet synchronous machine.
PM excited synchronous machines are more advantageous as compared with
electrically excited synchronous machine [32, 33].
The development of the PM machines has started due to the advent of ferrite PM
materials in 1950s, to enhance power density, efficiency, and compactness of
synchronous machines. Historical development of various magnetic materials with
their maximum energy product is as shown in Figure 1.7 [34]. With ferrite PM
10
material, these features could not achieve due to its low residual flux density and non-
linear demagnetization characteristics. However, the development in PM machine has
accelerated after the invention of high energy density Neodymium Iron Boron
(NdFeB) magnet in 1983.
Figure 1.7 Magnetic materials growth for the energy density (BH)max
The NdFeB magnets are produced by powder metallurgy process or by melt
spinning process. It consists of 65% Fe, 33% Nd and 1.2% boron [34]. AlNiCo and
SmCo are some other types of magnets that are also used in PM machine. NdFeB
magnets have high residual flux density and straight demagnetization curve as
compared to the other magnets. Table 1.2 shows the performance comparison of the
various PM materials [35].
Table 1.2 Magnet performance comparisons
Material Residual Flux
Density (Br)
Coercive Magnetic
Field Intensity (Hc)
Alnico 0.5-1.3 50-120
Ferrite 0.4 150-300
NdFeB 1.1-1.2 1000-2000
Samarium Cobalt 1.0-1.1 2000
Generally, the classification of the PM machines is according to the direction of
flow of flux, structure, type of motion, type of winding and type of core. The most
prominent classification of the PM machines is according to the direction of flow of
flux. There are three types of categorization regarding flux direction this includes
0
Ceramic magnet
NdFeB
Sm(Co, Fe, Cu, Zr)Z
SmCo5
Alnico
AlnicoMK Steel
KS Steel
(BH
)m
ax ( K
J/m
3) (B
H) m
ax (M
GO
e)
1920 1940 1960 1980 2000
Year
320
400
160
240
0
80
40
50
20
30
10
11
radial flux (RF), axial flux (AF) and transverse flux (TF) machines, as shown in
Figure 1.8 [36].
Figure 1.8 Classification of the machines according to the direction of the flow of
flux (a) RF machine (b) AF machine (c) TF machine
The radial flux PM machines are broadly classified into linear and rotary type
according to the direction of motion of the rotor. The rotary radial flux machines are
also classified into the rotor PM machine; having magnet on the rotor and the stator
PM machine; having magnet on the stator. Furthermore, rotor PM machines are
broadly categorized into inner rotor type and outer rotor type. Inner rotor types are
categorized into following main types, i.e. radial interior PM machine, circumferential
interior PM machine, surface PM machine and inset PM machine as shown in Figure
1.9 [37]. Outer rotor type PM machines are categorized into following types, i.e.
single bonded magnet ring type and magnet having p pole pairs as shown in Figure
1.10 [38]. The most common PM machines with magnet on the rotor are PM
brushless DC (PMBLDC) and PM brushless AC (PMBLAC) machine. The PMBLDC
12
machines have trapezoidal back EMF whereas PMBLAC machines have sinusoidal
back EMF. The PMBLDC machine is also known as a square wave machine. The
PMBLAC machines are called as either sine wave or PM synchronous machine. The
inner rotor PM machines predominate in the industry due to their outstanding
advantages [39].
Figure 1.9 Inner Rotor PM machine possibilities (a) Surface mounted (b) Surface
inset (c) Interior radial (d) Interior circumferential
Figure 1.10 Outer Rotor PM Machine variants (a) with single bond PM (b) with p
pole pairs
Mainly the categorization of the transverse-flux (TF) machine is based on the
configurations of PMs, phases and windings. The most common TF machines are of
the following types: surface-mounted PMs, flux-concentration PMs, axial-arranged
(a) (b)
(c) (d)
(a) (b)
13
phases, in-plane phases, double-sided windings and single-sided windings [40]. The
major disadvantages of the transverse-flux machines are complex magnetic circuit
construction, reduced power factor, large amount of the flux leakage and complex
manufacturing. In addition, TF machine are not very common in wind power
generation [39].
1.2 Significance of the thesis
This thesis presents an improved coreless AFPMSG's design, analysis and
optimization by using proposed magnet shape. The proposed coreless axial flux
machine presents a suitable alternative to meet increasing energy due to its reduced
cogging torque and torque ripple and increased output power and torque. The research
work presented in this thesis provides a comprehensive contribution in the field of
axial flux machine to be utilized for wind power applications.
A coreless AFPMSG benefits include high efficiency and low torque ripples than
core type AFPMSG. A DRSS coreless stator machine also eliminates the magnetic
unbalance. Furthermore, coreless AFPMSG is also suitable for the wind turbine
application due to its disc shape structure and high torque density. The design of the
AFPMSG is derived from the radial flux machine design. For the analysis and
optimization, a 2-D analytical method and a 3-D FEA are used.
Although the results of 3-D FEA are much closer to the experimental results,
however, it is very time consuming as compared to the analytical method. A 2-D
analytical method by using the mean radius approach is presented for the dual rotor
single coreless stator AFPMSG for the fast characteristics analysis. Furthermore, by
using the developed analytical method, optimization of the coreless AFPMSG is also
presented. The coreless AFPMSG optimum design by using the analytical method
reduces the time of optimization to less than a minute. Furthermore, in order to
confirm the effectiveness, the initial and optimized model results are also compared
with 3-D FEA.
Although, trapezoidal shaped PMs are most commonly used in disc shape
AFPMSGs. However, an arc shaped trapezoidal PM is proposed for decreasing
cogging torque and hence torque ripple. In addition, the proposed model optimization
is made to enhance the torque output and to reduce torque ripple further. A
comparison is made between the optimized and proposed models of the AFPMSG to
14
justify the proposed magnet shape. The volume of the PM is made constant for both
the conventional and proposed model.
1.3 Main contents of the thesis
To study the significance of the proposed PM shape AFPMSG, initially a dual
rotor axial flux machine is designed using the D3 method, instead of D
2L, which is
utilized in the radial flux permanent magnet machines. The machine parameters are
calculated, and a 1kW model was designed. For the rapid characteristics analysis a 2-
D analytical method is developed to see the parameters effects on the back EMF. The
rotor PMs shape was modified to an arc shape trapezoidal for decreasing cogging
torque and hence torque ripple. Then, the analysis by transient 3D FEA was done for
the performance evaluation of proposed magnet shape AFPMSG. For competitive
comparison, the developed machine performance is compared with its counterpart
trapezoidal shaped AFPMSG. Then, Kriging Method and Genetic Algorithm (GA) are
introduced and employed for the optimization of PMs. PM Overhang of rotor pole is
also considered as an optimization variable for improving torque density and
decreasing torque ripples of proposed machine. For competitive comparison, the
developed machine performance is compared with its basic model AFPMSG.
1.4 Thesis organization
The structure of this thesis as follows.
A brief description of the research background of the WECS is
presented in chapter 1. The description includes various types of
fixed and variable speed WECS and their comparison. The various
types of PMSMs are also discussed briefly. The contents and
significance of the thesis are also presented in chapter 1.
A brief description of the research background of the AFPM
machines is presented in chapter 2. The description includes various
types of the axial flux machine configuration and basic working
principle of the coreless AFPM machines.
In chapter 3, the design process of the AFPM machines is discussed
using D3 method. Electromagnetic design of the AF machine with
15
basic design equations is presented. The magnet operating point is
also discussed briefly. The 3D FEA approach is also discussed
along with Maxwell's equations.
In chapter 4, the importance of the AFPM machine analytical
analysis is discussed briefly. Furthermore, various analytical
methods developed for the characteristics analysis of the coreless
AFPM machine are also discussed. In addition, an optimal design of
an AFPM machine is presented by employing developed analytical
method.
In chapter 5, a novel arc shaped trapezoidal permanent magnet is
presented to enhance coreless AFPM machine performance.
Optimization of the AFPM machine having arc shaped PM is also
presented by employing asymmetric magnet overhang.
In chapter 6, the conclusion of the research work presented in this
thesis is presented.
16
Chapter 2 Axial Flux Permanent Magnet Machines
17
AFPM machines are inherently suitable for the direct drive application due to its
disc shape structure. They are usually more efficient because of the elimination of
rotor copper losses, high efficiency, compact size and reliability due to the absence of
external excitation. In some of the topologies of these machines, core losses are also
eliminated. Superior cooling characteristics and easy to manufacture are also main
advantages of various types of these machines. AFPM machines have high power and
torque densities. Furthermore, AFPM machines have a simple construction. Due to all
of these benefits AFPM machines are becoming a promising machine type for wind
turbine applications.
2.1 Axial Flux Permanent Magnet Machines Topologies
Davenport and Michael Faraday develop the initial radial flux and axial flux type
machines respectively. The patent published by the well-known scientist N. Tesla in
1889 was also about axial flux disc machine [41]. The advancement in axial flux
permanent magnet (AFPM) machines was slow due to its fabrication complexities
while comparing radial flux permanent magnet (RFPM) machines. This was mainly
due to the balance difficulty because of the very strong attracting stator and rotor
magnetic force [42, 43]. This was also due to the manufacturing of core for such type
of machines [44]. The development in the AFPM machines gained attention in the late
70s and early 80s. The rapid development of the AFPM machines initiated in 1980s,
owing to the growth in fabrication technology, as an alternative to the conventional
RFPM machines [45].
The most common applications of AFPM machines include hybrid electric vehicle
with flywheel energy storage, elevators, low and high speed wind power generation,
aircrafts, computer hard disk drives and vibration motors. The wind energy
technology is going popular very rapidly for becoming one of the most desirable
renewable energy source worldwide due to the depletion of the fossil fuels and
environmental friendly nature. The AFPM machines offer low cost as compared to
other solutions [46-53].
The single stator single-rotor (SSSR), double-stator single-rotor (DSSR), single-
stator double-rotor (SSDR), and multi-stator multi-rotor (MSMR) are the main
topologies of AFPM machines. These configurations are as shown in Figure 2.1 [54].
18
The AFPM machine are also classified according to the structure of PMs, winding
configurations and core type. The various such types are as follows: AF machine with
surface mounted or interior PMs, AF machine with armature slots or without armature
slots, AF machine with armature core or without armature core, AF machine with ring
winding or drum winding, AF machine with concentrated winding or distributed
winding, AF machine with integral slot or fractional slot winding and AF machine
with single layer winding- or multilayer winding [45]. Figure 2.2 shows a flow chart
of the various topologies of the AFPM machines.
Figure 2.1 AFPM Machine Topologies (a) SSSR (b) SSDR (c) DRSS (d) MSMR
The basic and simplest structure of the AFPM machine is SSSR. This axial flux
machine has generally slotted and slot-less type stator configurations as shown in
Figure 2.3 [55, 56]. The application of single sided AFPMMs is in gearless elevator,
servo electromechanical drives and in the military due to the compactness and high
torque density [57]. Main drawback of slotted single sided AF machine is the strong
unbalance magnetic force among rotor and stator cores. In order to overcome this
drawback either thrust bearing or the topology of single sided AF machine having
rotor or stator balance are used. Furthermore, the slot-less stator configuration of the
SSSR machine reduced strong magnetic unbalance among stator and rotor. Moreover,
SSSR machine has lower power density as compared to the double-sided AF machine.
(a) (b)
(c) (d)
19
Figure 2.2 Axial Flux Machine Topologies Flow Chart
Figure 2.3 SSSR Axial Flux Machine Topologies (a) Slotted type (b) Slot-less type
In DSSR AF machine having internal PM disc rotor, the armature or stator winding
is placed on both the stators. Figure 2.4 shows the slotted and slot-less topologies of
DSSR AFPM machines [45, 58]. It is also known as an axial flux inner rotor machine
(AFIR) or Kaman machine. The AFIR machine's rotor is located between either
slotted stators or slot-less stators. AFIR machine are categorized according to
arrangement of PMs as follows: surface mounted magnet type AFIR or buried magnet
type AFIR. The flow of flux in the surface mounted magnet type and inset magnet
type AFIR is along axial direction in rotor yoke. However, flow of flux in the buried
magnet type is along the circumferential direction. Furthermore, AFIR machine with
slotted stator is called a NS-type and with slot-less configuration is called either NN
or SS type [33].
Axial Flux Machines
Single Stator Single Rotor Double Stator Single Rotor Single Stator Double Rotor Multi Stator Multi Rotor
Iron Stator Core Iron Stator Core Iron Stator Core Iron-less Stator Core
Slotted
Stator
Slotted
Stator
Slotted
Stator
Iron Stator Core Iron-less Stator Core
Slotted
Stator
Slot-less
Stator
Slot-less
Stator
Slot-less
Stator
Slot-less
Stator
(a) (b)
20
Figure 2.4 DSSR Axial Flux Machine Topologies (a) Slot-less type (b) Slotted type
The winding of both the stators is connected either in series or in parallel.
However, the balance in the magnetic pull is the advantage of series connected type
AFIR machine. The main advantage of AFIR with parallel connection is its ability to
perform operation if one stator winding become faulty. The power density of surface
mounted PM type AFIR is greater as compared to interior PM type AFIR due to its
thinner rotor back iron requirement. Also the power density of the slotted type AFIR
machine is lower than the slot-less type machine. However, the slot-less type AFIR
has low efficiency due to high copper losses because of the increased end windings.
The armature reaction and PM ends leakage flux in interior magnet type AFIR is
higher while comparing with surface-mounted magnet type AFIR. However, the
interior PM type AFIR machine protects better against magnet mechanical force, wear
and tear, and oxidization [45, 59].
The DRSS AFPM machines consist of armature winding sandwiched between two-
rotor discs [60]. Figure 2.5 shows the most common configurations of DRSS AFPM
machine [45]. The stator of double rotor single stator type machine is either core-type
or coreless type. Furthermore, the core type stator can be either slot-less or slotted
type [61, 62]. Both SSDR and DRSS AFPM machines are also named as three disc
machines. In addition, the rotor of the DRSS machine is surface mounted PM, inset
PM or buried PM types [63-65].
In the DRSS coreless AFPM machine, flow of flux is in axial direction. However,
flow of magnetic flux in the DRSS core type AFPM machine is along circumferential
or axial direction. The flux path in the various DRSS AFPM machines are shown in
Figure 2.6 [66]. The coreless AFPM machines are of NS-type. However, the core type
(a) (b)
21
AFPM machines are either NS or NN type [45]. Shapes of the magnets that are most
commonly employed in DRSS AF machines are as shown in Figure 2.7 [67].
Additionally, the DRSS AF machine is also found in literature with Halbach
magnetization. The rotor back irons are removed by using this type of an arrangement
[61].
The Coreless AFPM machine has higher efficiency, as compared to the core AFPM
machine [68]. Furthermore, the dual rotor coreless stator AFPM machine has a
reduced total harmonic distortion (THD) in the back EMF. The high efficiency and
more sinusoidal back EMF are due to the absence of stator core. Moreover, this
machine has greater mechanical stability due to the reduced axial force of attraction
between the stator and rotor, and it is also easy to manufacture [69-71].
Figure 2.5 DRSS AFPM machine Topologies (a) Coreless type (b) Slotted type (c)
Slot-less type
(a)
(b)
(c)
22
Figure 2.6 Flux path in DRSS topologies (a) NN Slot-less Core type (b) NS Slotted
core type (c) NS coreless type
Figure 2.7 Magnet shapes in DRSS AFPM machines: (a) Trapezoidal (b) circular (c)
semi-circular
In multistage AFPM machine, there are multiple discs of the stator and rotor. It is
also known as multi-stack or multidisc AFPM machine. The simplest multistage
configuration is the double stage configuration. Double stage configuration has either
two-stator disc with three rotor discs or two-rotor disc with three stators. The multi
A C B A C
A C B A C
N S
S N
φ
Rotor Back Iron
Rotor Back Iron
Stator Yoke
φ
φ
φ
φ
A C B A C
A C B A C
N S
N S
φ
Rotor Back Iron
Rotor Back Iron
Stator Yokeφ
(a)
(b)
(c)
N S
Rotor Back Iron
φ φA A C C
N S
Rotor Back Iron
B B A A C C BB
(a) (b) (c)
23
disc machine is made from either DRSS or SRDS topologies. In a multidisc machine
with DRSS, the number of stator disc are one less than the rotor disc. However, in a
multidisc machine with SRDS, the stator discs are one more than the rotor discs.
Multidisc machine is used where it is not feasible to increase the power rating of the
SRDS and DRSS machines due to the mechanical constraints. The various
configurations of the multidisc machine exist as in the case of DRSS and SRDS.
Figure 2.8 shows a configuration of the multistage axial flux machine [45]. Multidisc
machines are used in high speed PM generator, pumps, and ship propulsion [36, 45,
72].
Figure 2.8 Multistack or multistage AFPM machine
2.2 Winding configuration in AFPM machines
There are two main classifications of the winding in the electric machine, i.e.
overlap and non-overlap winding as shown in Figure 2.9 [73]. Overlap winding type
can either concentrated or distributed. The concentrated overlap winding is having
one slot/pole/phase. However, the distributed overlap winding is having more than
slot/pole/phase. On the other hand, non-overlap winding is of the concentrated
winding type. The categorization of the non-overlap winding is into single and double
layer winding. The number of coil sides per slot in a single layer winding is one. In
addition, number of stator slots are double to the number of armature coils in single
layer winding. Whereas, double-layer machine winding is having two winding coil
sides in each stator slot. Non-overlap windings are also termed as fractional slot
concentrated windings. Machine windings with slot/pole/phase≤1 are termed as
integral or fractional slot concentrated windings. The classification of the winding
according to coil pitch is termed as full or fractional pitch winding. The winding in
which the coil and pole pitches are equal is termed as full pitch winding. However,
the winding in which pole pitch is greater than the coil pitch is termed as fractional
24
pitch winding. The back EMF of coil sides are additive and have no phase difference
in full pitch windings. In fractional pitch winding, coil sides do not have a zero phase
difference. There also exists the categorization according to the winding around the
teeth or core. If it is around the core, it is a drum winding and if it is around the tooth,
it is a ring winding [59, 74-76].
Figure 2.9 Typical winding configurations (a) Overlapping (distributed) (b)
Overlapping (concentrated). (c) Nonoverlapping, all teeth wounds (d)
Nonoverlapping, alternate teeth wound.
Winding of the AFPMM is either overlap or non-overlap and their sub types
include concentrated or distributed single layer and double layer windings.
Concentrated type has the benefit of easy to manufacture, low copper losses, cost
reduction because the size of copper and the axial length of machine decreases.
Distributed winding has the benefit of more sinusoidal back EMF on the expense of
increase in manufacturing cost and power loss due to increase of end turns length of
winding. The core type dual rotor is of either teeth wound or core wound type with
concentrated or distributed winding as shown in Figure 2.10 [59]. However, the
winding of the coreless dual rotor machine is of ring type. Furthermore, the coreless
A12
A11
C12
C11
B11B12 B1
A1
C1
B1
C1
A1
C1
A1
B1
(a) (b)
(c) (d)
25
type machine has either single layer or multilayer winding with both concentrated and
wave winding configurations as shown in Figure 2.11 [59, 69, 77-79].
Figure 2.10 Winding types in core type structure of AFPM machines (a) Tooth
Wound (ring type) (b) Core Wound (drum type) (c) Ring Type Concentrated (d)
Drum Type Concentrated (e) Drum Type Distributed
Figure 2.11 Winding types in coreless structure of AFPM machines (a) Single Layer
Concentrated (b) Double Layer Concentrated (c) Triple Layer Concentrated (d) Triple
Layer Wave Winding
(a) (b)
(c) (d) (e)
(a)
(c)
(b)
(d)
26
2.3 Summary
In this chapter, a review of the various AFPM machine configurations is discussed.
The coreless DRSS AFPM machine exhibits increased efficiency and reduced torque
ripples as compared to the other AFPM machines. Furthermore, various
configurations of the coils and magnets in the coreless AFPM machines are also
discussed. Trapezoidal shaped coils and magnet are commonly used in the AFPM
machines due to its increased output torque capability as compared to other shape.
27
Chapter 3 Design and Analysis Procedure
28
The coreless AFPMSG has the benefit of high efficiency. This is achieved due to
the elimination stator core losses. It has also the advantage of reduced cogging torque
and hence more suitable for low speed operation. Furthermore, it also eliminates
imbalance axial force on the stator and thus provides smooth operation. The design of
coreless stator AFPM machine having dual rotor discs is presented in this chapter. In
this configuration coreless stator is sandwiched between twin external rotors.
Furthermore, PMs are fixed on the rotor back iron surface in this configuration. First,
PM operating point and permeance coefficient are discussed briefly. The basic
electromagnetic design equations of the AFPM machine are developed in this chapter.
The design process of AFPM machine utilizing D3 sizing procedure is also presented.
The presented D3 sizing method is similar to the D
2L sizing method which is used in
designing RFPM machines. Moreover, the 3D-FEA approach is discussed along with
Maxwell's equations.
3.1 Magnet Operating Point
Generally, the permanent magnets are characterized by having a large hysteresis
loop. A hysteresis loop is formed by switching on and off the field intensity in a non
magnetized material. The magnet operating point is influenced by the magnetic
environment in which the magnet is placed. The point of an intersection of a load line
and PM demagnetization characteristics is called as magnet operating point and is
shown in Figure 3.1 [80]. The permeance coefficient (PC) is obtained by calculating
the slope of the load line. Along the horizontal axis, PC is zero and along the vertical
axis, its value is infinite. However, the value of the PC, vary between these two
extremes depending on the magnetic environment or the permeance [81].
To calculate the magnet operating point, let us consider an infinite permeable rotor
and stator cores in a magnetic circuit, as shown in Figure 3.2 [80]. By using ampere's
law MMF of magnetic circuit is given by the equation (3.1) [82].
(3.1)
Where Hm is magnet region field intensity, Hg is air gap region field intensity, hm is
magnet length and lg is air gap region length.
29
Figure 3.1 Operating Point of the Magnet
Figure 3.2 Basic Magnetic Circuit
A relationship between air region flux density (Bg) and air region magnetic field
intensity (Hg) is given by the equation (3.2).
(3.2)
Where µo is free space permeability.
The leakage flux can be neglected by considering the magnet operating flux equal
to an air region flux. A relationship between magnet region and air region flux
densities is given by equation (3.3) by using Gauss's law while neglecting leakage
flux.
(3.3)
Where Bm is the magnet operating flux density, Ag is an air gap region cross-
sectional area and Am is magnet cross-sectional area.
The PC is also stated the ratio between Bm and µoHm on the load line. The PC is
given by equation (3.4).
(3.4)
-µ0Hc µ0Hm µ0H
B
Br
Bm
µR
PC
Magnetic Field Intensity
Ma
gn
etic
Flu
x D
ensi
ty
N S
Rotor Back Iron
Stator Yoke
lg
hm
φ
30
The relationship between B and H on the demagnetization characteristics is given
by the equation (3.5).
(3.5)
Where Br is magnet residual or remanent flux density and µrm is the magnet relative
permeability.
By substituting Equation (3.1) and (3.3) into equation (3.5), Bm is derived as by
the equation (3.6).
(3.6)
This shows that the Bm is smaller than the Br and it is dependent on the permeance
coefficient. The calculation for the length of the magnet is obtained by using equation
(3.4) for any designed value of the PC. This equation also provides design guidelines
regarding the various magnet dimensions to calculate Bm. Similarly, to calculate the
operating flux density at various loads, a left shifted air gap line has to be drawn by an
amount equal to an external magnetic field intensity. An intersection of such a load
line with the demagnetization characteristics determines the operating flux density at
that load current.
3.2 Design Method
The frequency and speed of a machine determine the number of the pole. The
frequency and pole pair for rotational speed in revolution per min and revolution per
second are related by the equations (3.7) and (3.8) respectively.
(3.7)
(3.8)
Where Ω is mechanical speed in revolution per minute (RPM), ns is the mechanical
speed in revolution per second (rps), f is frequency in cycles per second, P is number
of machine poles and p is number of machine pole pair.
Also, the relationship between frequency and speed for the quantities in radians is
as given by the equation (3.9).
31
(3.9)
Where ωr is mechanical speed in radian per second and ωe is an electrical speed
(magnetic or electric field) in radian per second of the machine.
For the single phase operation of the machine, number of coils are either equal to
number of poles or multiple of poles. For the multi phase operation of the machine, a
suitable combination of the coils and pole numbers is selected as given by the
equation (3.10) [83].
(3.10)
Here n is a constant, m represents number of phases and S represents stator coils
count. The ratio between n and m must be a non-integral value to obtain feasible coils
and poles combination.
In dual rotor with internal stator axial flux machines, the coreless, slot-less or
slotted type armature windings are used. The coils of the coreless type armature
winding are made rigid by utilizing epoxy resin in the coreless AFPM machines.
Generally, the coil profile used in the coreless winding is of trapezoidal, rhomboidal
and circular shapes. Trapezoidal coils have the advantages of increased torque density
as compared to other profiles [67]. Furthermore, the magnet profiles that are used in
the coreless AFPM machines are of trapezoidal, circular or semi-circular shapes. The
various relations used in the coreless winding are as follows.
In a coreless configuration, concentrated type single layer winding is generally
utilized. If number of coils at stator (S), is known, then coils per phase (Cn), coils per
poles (Q) and coils per poles per phase (q1) are given by equations (3.11), (3.12) and
(3.13) respectively [67].
(3.11)
(3.12)
(3.13)
If Tph is number of turns in a phase, then turns per coils (Nc) and conductors per
coils (nc) are calculated by using the equations (3.14) and (3.15) respectively.
32
(3.14)
(3.15)
Where aw is number of parallel conductors.
Parallel conductors or Litz wires are generally used in coreless AFPM machines to
reduce copper losses or joules losses [84]. The inner, outer and average coil pitches of
the disc type machine are calculated by using the equations (3.16), (3.17) and (3.18)
respectively. Whereas the inner, outer and average poles pitches are given by the
equations (3.19), (3.20) and (3.21) respectively. Inner, outer and mean pitches
calculations are presented because we are considering trapezoidal shaped coils and
magnets in the configuration of the coreless AFPM machines.
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
Where Di, Do and Da are the rotor disc inner, outer and average diameters. The
rotor disc average diameter is calculated by the equation (3.22).
(3.22)
For the AFPM machines, in order to simply the equations ratio between internal
and external rotor diameters is written as kd, given by the equation (3.23).
(3.23)
33
The inner and outer pole arc widths are calculated by using the values of the
calculated pole pitches and by assuming an appropriate pole arc to pole pitch ratio.
Also, the radial length or thickness of magnet is calculated by using the equation
(3.24).
(3.24)
The back iron length is calculated by using equation (3.25).
(3.25)
Where, Bmax is the maximum allowable rotor back iron flux density.
The other significant parameter is stack length that is calculated by using the
equation (3.26).
(3.26)
Whereas, Ly is yoke height, Lw coil length in axial direction, lg is the air gap length,
hm is pole length in axial direction.
Three phase winding factor kw is obtained by multiplying distribution factor kd1 and
pitch factor kp. Both these winding factors influence the back EMF. The values of
both these factors are obtained by using the equations (3.27) and (3.28) respectively
[85].
(3.27)
(3.28)
Where β is a ratio between coil and pole pitches and it is given by the equation
(3.29).
(3.29)
Where τpa is an average pole pitch and τca is an average coil pitch.
The electrical loading Am is considered as number of ampere conductor around the
air gap circumference. Its peak value is written by the equation (3.30) [86].
34
(3.30)
Let us consider a sinusoidally distributed flux density due to PM. Its average flux
density is given by the equation (3.31).
(3.31)
The flux per pole due to the circumferential element per pole i.e. 2πrdr/P is given
by the equation(3.32).
(3.32)
Where Bm is maximum flux density and α is ratio between averages and peak
magnetic flux densities
By substituting Do=0.5Ri and by using equation (3.23), the flux per pole is given
by the equation(3.33).
(3.33)
For a phase winding with winding factor kw, having Tph turns placed in a sinusoidal
distributed magnetic flux density having flux per pole Φp, generally the flux linkage at
any time instant t is given by the equation (3.34).
(3.34)
Since
(3.35)
By using equation (3.35), flux linkage can be expressed by the equation(3.36).
(3.36)
By using Lenz law, the back EMF's rms and peak values are obtained as given by
equations (3.37) and (3.38) .
(3.37)
35
(3.38)
The output power in the single stator axial flux machine is given by the equation
(3.39).
(3.39)
By using equations (3.30)and (3.38), the output power is as given by the equation
(3.40).
(3.40)
The torque can be found be using equation (3.41)
(3.41)
The rotor outer diameter is calculated by the equation (3.42) by solving equation
(3.40).
(3.42)
The calculation of the efficiency is as given by the equation (3.43).
(3.43)
The output torque of a RFPM machine is directly proportional to stack length and
rotor diameter square. However, for an AFPM machine output torque is directly
proportional to rotor diameter cube as it can be seen from the equation (3.43). This
shows that axial flux machine produces more output torque for the same rotor
diameter as compared to the radial flux machine.
Generally, size of an electrical machine is determined by its torque per unit rotor
volume (TRV). The TRV is electric and magnetic loading product and it is given by
the equation (3.44) [86].
(3.44)
Where Vr is the volume of the rotor and Bm is magnetic loading.
36
Torque is also related with the average shear stress σ at the rotor surface [87]. The
shear stress is force per unit area. In electric machines shear stress produces torque.
The torque in terms of shear stress is as given by the equation (3.45).
(3.45)
By using equations (3.44) and (3.45), the TRV in terms of shear stress is as given
by the equation (3.46).
(3.46)
In the electric machines, the type of the material used limits both the electrical and
magnetic loading. The electrical loading Am is linear current density along the
circumference of air gap. The Am measure how many amperes packed inside each unit
of stator circumference. The various cooling methods limit electrical loading i.e.
totally enclosed, fan cooled and liquid cooled. The Bm is taken as an average flux
density over rotor surface and it is usually sinusoidally distributed. Table 3.1 shows
typical values of TRV for the various categories of PM machines [86].
It is very useful to relate electrical loading with the current density. The electrical
loading relation with current density J is given by the equation (3.47).
(3.47)
Table 3.1 Standard values for TRV
Types of machine TRV
KNm/m3
Totally enclosed machines (with ferrite PM) 7~14
Totally enclosed machines (with rare earth PM) 14~42
Integral-hp industrial machines 7~30
Table 3.2 shows the various cooling conditions related to electrical loading and
current density in PM machines [86]. Typical value of magnetic loading for an air-
cooled PM machines is around 0.7T. However, it is around 0.8 T for the liquid cooled
37
machines. Electric machines are usually designed to get high torque and power
densities by considering the maximum limits of the current and magnetic loadings.
Higher electric loading causes overheating in the coils and PMs. Demagnetization of
the PMs occurs with excess heating. Therefore, current loading is kept below the
permissible limit in order to avoid damage. The magnetic loading limit is made to
avoid the back iron magnetic saturation.
Table 3.2 Selection of electrical loading and current density
Conditions Electric loading Current density
A/mm A/mm2
Totally enclosed Around 150 1.5~5
Fan cooled Around 350 5~10
Liquid cooled Around 600 10~30
Based on the assumed value of the shear stress it is possible to get an approximate
value of the outer diameter after computing rotor volume. It is also possible to get an
approximate value of the outer diameter by assuming current and magnetic loading.
The rotor disc inner diameter is found by assuming of inner and outer diameter ratio
for the maximum output power .i.e., . The proper selection of coils and poles
combination is dependent on winding factor. A flow chart of the design process is
shown in Figure 3.3.
Figure 3.3 Flow chart of the design process.
Rotor yoke thickness
Rated power, Rated voltage,
Frequency, Synchronous speed, Power
factor, and Efficiency
No. of Poles and Coils
Winding factor
Magnetic and electric loading
Rotor outer and inner diameter
Pole and coil pitch
Flux per pole
Conductor cross sectional area
Current Density
Coil height
Total stack length
Induced voltage
Magnetic pole thickness
Conductor per coil
No. of coils per phaseFull-Load current
No. of turns per phase
Torque, Power
38
3.3 Analysis Method
For the analysis of electric machines, various traditional solutions, i.e. magnetic
equivalent circuit and analytical solutions of the Poisson and Laplace equations are
used. The magnetic equivalent circuit method is fast and it is a fine starting point. Its
main drawback is the considering of lumped parameter and ignoring magnetic flux
density spatial distribution. However, an analytical method considers these effects in
the computation of magnetic field distribution, but it could not deal with complex
shapes and saturation effects. Both these methods provide rapid performance analysis
and deep insight into the machine design by utilizing magnetic circuit theory.
However, due to the highly nonlinear fields, magnetic cross coupling, complex
geometry and saturation effects in the electric machines, the conventional analytical
field solutions do not give satisfactory results. Therefore, to provide correct
performance prediction, numerical techniques like Finite Element Method (FEM),
Boundary Element Method (BEM) and Finite Difference Method (FDM) are utilized.
FEM provides an accurate estimate of the design by considering all the nonlinear
fields created. Results are obtained in this research work by using JMAG Designer (a
commercially available FEM solver).
Although, the structure of the electric machine is 3D and a 3D-FEM is required for
its performance analysis. However, in radial flux machine, both the two dimensional
finite element analysis (2D-FEA) and (3D-FEA) are generally used. In 2D-FEA for
radial flux machine, the field problem is reduced to the 2D due to its symmetry or 2D
electromagnetic problem nature. This is because 2D-FEA is preferred for the rapid
performance analysis. For the rapid performance analysis of the radial flux machine,
the half or quarter model is also utilized due to its symmetrical electromagnetic
nature. To study axial flux machines magnetic field, the 3D-FEM is utilized due to the
nature of its tri-dimensional electromagnetic problem.
To obtain the solution by the FEM, geometry of the model is discretised into
elementary elements, known as "finite elements". The most commonly used elements
in 2D-FEM and 3D-FEM are quadrilateral and tetrahedral. The assembly of various
elements is called as mesh. The distribution of the magnetic potential is presented by
a partial differential equations obtained from Maxwell's equations. Within each
element, the magnetic vector potential is approximated to vary according to shape
functions. In FEM, the quality of mesh is very important for the precision of the
39
performance analysis. The commonly employed meshes in JMAG designer are
automatic, adaptive, slide and rotate, layered, thin plate, skin depth, patch, spatial
automatic and manual. The value of field quantities at any points inside each element
is interpolated from the vertices of the element.
To solve the electromagnetic field problem, the governing Maxwell's equations are
required. For the sake of simplicity partial differential Maxwell's equations are written
in magnetic vector potential form. Generally numerical techniques are applied in
Maxwell's equation solution.
The magnetic field intensity is related to scalar magnetic potential (U) as given by
the equation 3.48 in the airspace region.
(3.48)
(3.49)
(3.50)
(3.51)
The magneto-static problem is described by the following equations.
(3.52)
(3.53)
(3.54)
Since, ∇ .(∇ ×B)=0 , the magnetic vector potential is defined as.
(3.55)
Here H is a representation of magnetic field intensity, B is a representation of
magnetic flux density, µ is a representation of magnetic permeability, Ji is a
representation of current density, ∇ is an operator while A is a magnetic vector
potential. ∇ is called as del or nabla and it is used to reduce length of partial
differential equation. By considering a 3D magnetic vector potential A3D for a 3D-
FEM problem, the equation 3.52 can be written as follows by using the equations
3.53 and 3.55.
40
(3.56)
Since
(3.57)
Equation 3.57 is expressed as given by the equation 3.58 by considering Coulomb
gauge law i.e. ∇ .
(3.58)
This results into Poisson's equation as given below by using equation 3.56.
(3.59)
With A3D in terms of the unit vectors (eρ, eφ, ez), Poisson's equation is
expressed as given by the equation 3.60, in the cylindrical coordinate system (ρ,
φ, z).
(3.60)
Where, each components of the Laplacian operator are given as below.
(3.61)
(3.62)
(3.63)
In the numerical techniques, the quality of meshing is very important which
determines the analysis precision mesh. There are numerous possible mesh
approaches for meshing the air regions between stators and movers in commercial
FEM software JMAG- designer. However in this research work re-meshing at each
step is used due to its numerous benefits over other meshing techniques [88].
41
3.4 Optimization Method
Subject to equality and inequality constraints the optimal design problem is
converted to the minimization of an objective function. The optimum design problem
is to get a point in the feasible domain for minimizing an objective function. The
optimal design problem is given as [89]:
For minimizing cost function f(x) conditional on inequality constraint gi(x) ≤ 0,
i=1 to m as well as equality constraint hj(x)=0, j=1 to p, find an optimal design
variable vector x.
The feasible set for design problem is an accumulation of all feasible designs.
The feasible set S for a design problem of all feasible designs can be stated as
follows:
(3.64)
Global (Absolute) Minimum
At x* an objective function f(x) has a global minimum when inequality f(x*)≤ f(x)
satisfies entire x in the feasible set S.
Local (Relative) Minimum
At x* an objective function f(x) has a local minimum when inequality f(x*)≤ f(x)
satisfies all x in a small neighborhood N (vicinity) of x* in the feasible set S.
Consider a one variable function f(x) to identify graphical meaning of local and
global minima as shown in Figure 3.4 [89]. In this waveform, the local minima of this
function are points B and D, because the minimum value of this function lies in
vicinity of these points. Likewise, local maxima of this function are points A and C.
The value of x and f (x) may lie between - ∞ and ∞, thus both domain and function are
boundless. Therefore, the f(x) has no global maximum or minimum. The function's
global maximum is at point F whereas, function's global minimum is at point E when
x is limited between -a and b, as can be seen in Figure 3.4(b). Here, points E as well
as F comprise bounded constraints, whereas A, B, C, and D are unbounded
constraints.
A point is called as a global minimum if it has no other promising points with
improved values of the cost function.
A point is called as a local minimum if it has no other promising points "in
the surrounding" with improved cost function values.
42
Figure 3.4 Optimal points: (a) The boundless domain and function (local minima and
maxima), (b) The bounded domain and function (global minima and maxima)
By using local optimization techniques, which are conventionally used, the
optimization of electrical machine design is considered as a multi-variable constrained
nonlinear problem. The selection of initial points may result in an undesirable local
solution by using the conventional optimization methods. The general solution for
these techniques drawbacks is problem solving by taking different initial points. For
the design of electric machine, the number of design variables should be minimum to
avoid complexity. Thus, the design problem of electrical machine is phrased to be a
global constrained problem [90, 91]. An optimization method considering the Latin
Hypercube Sampling (LHS), Kriging method and Genetic Algorithm (GA) has been
employed for electrical machine optimization [92]. For the estimation of mean,
deviations and distribution of functions LHS is considered for accuracy as compared
to random and stratified sampling in constructing Kriging model. Furthermore, this
guarantees that all design variables represents all its range [93]. In addition this can be
a useful tool to optimize a machine design if samples are selected in moderation i.e.
not too far from design point or two points are not very close to each other. In this
research, for employing the LHS a MATLAB Model-Based Calibration Toolbox is
used.
3.4.1 Kriging Method
The discussion presented below regarding Kriging method is referred to [94, 95].
The Kriging model is built by the geostatistics method and maximum likelihood
estimation method for interpolating the random field values. It is a commonly used for
the calculation of the minimum error variance. To minimize error variance in the
(a) (b)
43
Kriging model, the bias is eliminated by using estimated equation. The Kriging model
is stated as given below:
(3.65)
Where f(x), y(x) and z(x) are the approximation function, interpolation function and
stochastic process realization function respectively. The z(x) matrix of covariance is
stated as given below:
(3.66)
Where R is the correlation matrix and and R(xi, xj) is the correlation function
respectively between points xi and xj. The various numbers of the correlation function
which exist are specified by the user. The selected Gaussian correlation function is
written as follows:
(3.67)
Where ndv is total design variables, θk and pk are undetermined correlation variables
for fitting the model, and xki, xk
j are sample points components x
i and x
j of the kth
component.
The y^(x), as an estimated value at x is written as follows by using relationships of
correlation among θk and best linear unbiased prediction of the y(x).
(3.68)
Where is the regression coefficient for the mean square predictor, the correlation
between sample points x and n is equal to r(x)
(3.69)
The Kriging modeling is determined based on the θ=[θ1......θndv] and
P=[p1......pndv] correlation elements. By the help of estimation by maximum
likelihood a prime predictor is determined. To be precise, by maximizing the
likelihood function the correlation elements are determined as given below
(3.70)
Where maximum likelihood prediction value σ^2 is as follows:
44
(3.71)
3.4.2 Genetic Algorithm
As discussed in [96], the GA is a heuristic search algorithm which draws its
analogy from the nature. The GA is generally applied to produce valuable solutions
for optimization and the search tasks. This algorithm is used for solving both the
bounded and unbounded optimization problems. It solves optimization problems
using natural evolution techniques, i.e., crossover, metamorphosis, and selection. GA
does not depend on initial point of search. In addition, it does not use functional
derivative information or any other auxiliary information of objective function. Its
trapping chance is lowest in local minimum.
A flow chart of the GA is as shown in Figure 3.5[96], chromosomes which encode
individuals to optimization problem, go forward toward better solutions. Binary code
of 0's and 1's are used to represent the solutions, furthermore other types of encoding
can also be used. An evolutionary process usually begins by forming a randomly
created individuals population and occur in the generations. Every candidate fitness
inside each generation is assessed, multiple candidate solutions are chosen from
present population (depending on its fitness), and for making fresh population they
are altered by regrouping and randomly mutating. The fresh population is then chosen
in a subsequently cycle of methods. Generally, the algorithm ends if an appropriate
fitness standard or the creation of the maximum number of generations is obtained.
By GA, the coupling effects of design parameters are taken into consideration by
simultaneously optimizing entire populations of designs at once rather than a single
design at a time.
Furthermore, GAs for the distinct objective optimization problems can also be used
for solving various objective optimization problems. In addition, the most familiar
technique for the multi-objective optimization problem is to use weighted sum
method, and it is realized by
(3.72)
Here, w is weights vector set by the decision maker i.e., and w > 0. If
any objective is not normalized, we need not add to 1.
45
In general, for this combined optimization process, the LHS is applied in order to
select the sampling points using design of experiment. In addition the Kriging method
is utilized for model approximation. Furthermore, a genetic algorithm is made in
achieving optimal values for the design variables.
Figure 3.5 The flow chart of GA.
3.5 Summary
In this chapter, a three phase coreless AFPM machine design is discussed by
developing sizing equation which possess the advantages such as high efficiency and
power density. The machine design included comprehensive calculations equation for
the various parameters. Furthermore, approach of the machine design by using sizing
equation is also discussed. The design parameters of the AFPMSG are presented in
chapter 3 and chapter 4 by utilizing electromagnetic design equations presented in this
chapter. In addition, transient 3D FEA approach is also presented along with
Maxwell's equations. Transient 3D FEA is utilized for the design and analysis of the
AFPMSG in chapter 3 and chapter 4. Also the optimization method by using the
Krigging Method and GA is discussed. The optimization of AFPMSG is presented in
chapter 4 by utilizing Krigging method and GA.
Breed new individuals through crosser and mutation operation to give
birth to offspring
Evaluate the fitness of new individuals
Replace least fit population with new individuals
Start
Repeat on this generation until termination
(time limit, sufficient fitness achieved, etc)
Select the best fit individuals for reproduction
Evaluate the fitness of each individual in the population
Choose the initial population of individual
End
46
Chapter 4 Analysis of AFPMSG with 2-D Analytical
Method
47
4.1 Introduction
For accurate computation of machine characteristics, e.g., output torque and back
EMF, the precise calculation of an air gap flux density is essential. The magnetic
equivalent circuit method, an analytical method and FEA are used for computation of
magnetic field. Among these methods FEA is more accurate, since others methods
can not tackle nonlinear magnetic fields. The disadvantage of using FEA is that it is
very time consuming and hence the effect of varying various parameters on the output
is very time consuming [68, 97]. Therefore, analytical methods are favored for finding
a time-efficient solution.
The PM machines, magnetic field computation is a focus of recent research
nowadays. The magnetic flux distribution in a PM machine is computed either by
using a polar or rectangular coordinate system [98, 99]. The computation of the
magnetic field for the double sided slot-less stator axial flux machine is made by
representing magnets and coils as current sheets in [100]. An analytical method of the
normal field component was developed for the single stator and rotor axial flux
machine with flat multiple pole disc magnets by using the mean radius approach
[101]. The distribution of the magnetic field for the single sided coreless stator axial
flux PM generator has also been computed using the mean radius approach [102]. A
voltage total harmonic distortion (VTHD) calculation of the single-sided coreless
AFPMSG using a 2-D analytical method is presented in [102]. An optimization for
reduction in VTHD of the double sided AFPMSG using 3-D FEA is presented in
[103]. However, an analytical solution to magnetic field distribution of coreless dual
rotor single stator AFPMSG has not been presented yet, which uses the mean radius
approach.
This chapter presents a 2-D analytical method to calculate back EMF of coreless
AFPMSG with dual rotor. The developed analytical technique is further expansion of
analytical method developed in [102]. Analytical method presented in [102] is for the
single rotor and coreless stator whereas the developed analytical method is for dual
rotor and coreless stator. The equations are developed by using the same approach i.e.
mean radius, rectangular coordinate system and solution of the Maxwell's equations
as was presented in [102]. However, there is a minor difference in the derived
equations and the equations that were presented in reference [102] due to the different
position of lower and upper magnets from the reference point. In the developed
48
analytical method equations are not being simply taken from the references [100-102]
for the superposition but they are derived. Furthermore, developed analytical method
is used for the optimization of the dual rotor and coreless stator in this paper instead if
time consuming 3D FEA. A transient 3-D FEA is employed to verify results of the
developed analytical method for both initial and optimized models. JMAG designer
ver. 14.1 is used as a 3D-FEA tool in this chapter. Performance evaluation of initial
and optimized model of the AFPMSG under no load and load conditions is done with
time stepping FEA, since it is more accurate as compared to the magneto-static FEA
for the rotating electric machines. Finally, the VTHD of the initial and optimized
models is compared using 3-D FEA.
The rest of chapter is arranged as follows: Section 4.2 presents 2-D analytical
method for the modeling and analysis of coreless AFPMSGs. This is followed by
Section 4.3, which describes the optimization of the AFPMSG using the developed
analytical method. In Section 4.4, a summary is drawn.
4.2 2-D Analytical Modeling for Coreless AFPMSG Analysis
This section presents 2-D analytical method for calculating the magnetic flux
density with Maxwell’s equations using boundary conditions of the single coreless
stator and dual rotor AFPMSG.
4.2.1 Initial Model
Figure 4.1 shows the initial model of the 1.0 kW, three phase, Y-connected,
double-sided AFPMSG. The AFPMSG has two disc-shaped rotor yokes with
permanent magnets placed on it. The coreless stator is sandwiched between two rotors
and has stator windings fixed by the plastic resin. Three phases of the stator windings
are arrayed periodically in the circumferential direction. Table 4.1 presents various
design parameters of the AFPMSG for 2-D analytical method.
4.2.2 Assumptions
The analytical model is developed for a double sided AFPMSG with coreless
armature winding. The following assumptions are made for the sake of ease.
i. There is no magnetic saturation and rotor discs have infinite permeability.
49
ii. The PMs have uniform magnetization. Furthermore demagnetization of a
magnet is a straight line. It has a constant relative permeability and its value is
close to unity.
iii. While calculating no-load magnetic field of upper region PMs, lower regions
PMs are considered as free space and vice versa.
iv. The whole magnetic regions are taken as air region or free space for armature
reaction field computation.
While developing analytical model, the no load and armature reaction magnetic
fields are computed independently because of magnetic linearity. The computed no
load and armature reaction fields are merged for obtaining resultant magnetic field by
using superposition.
Figure 4.1 Exploded view of the 3-D FEA model of the AFPMSG.
Table 4.1 The AFPMSG parameters
Parameters Units Values
Outer radius of rotor mm 81.2
Inner radius of rotor mm 54.8
Height of back iron core mm 4.0
Interpolar separation mm 13.23
Height of magnet mm 10.0
Height of machine mm 46.0
Air gap mm 1.5
Pole Pitch mm 35.73
Speed rpm 1100
No. of poles - 24
Turns/phase - 396
No. of coils - 9
Coil phase a
Coil phase b
Coil phase c
Rotor back iron
PM South PM North
50
4.2.3 Magnetization of the PMs
The relation among vector fields B and H for air region and magnet region is
as follows [98].
air region (4.1)
magnet region (4.2)
Where M is magnet residual magnetization vector, the
permeability and is relative permeability of magnet.
In scalar magnetic potential form, we can write the following relation as
below [98].
(4.3)
and
air region (4.4)
magnet region (4.5)
The magnetization vector is towards axial direction and in Cartesian
coordinates the magnetization vector is given as follows.
(4.6)
Where
Therefore, equation (4.6) reduces as follows:
Figure 4.2 Magnetization produced by the PMs.
(4.7)
is obtained by taking the Fourier series of the Figure 4.2. The general
form of the Fourier series of the any even function is as [98].
(4.8)
Where
M
2
m2
m
p2
mp
0
rB
x0
51
(4.9)
Where L is the period of the function, which is in this case.
Now in order to calculate , by using equation (4.9) we can write as
follows:
(4.10)
By solving the equation (4.10) we can write as follows:
(4.11)
Now in order to calculate , again by using the equation (4.9) we can
write as follows:
(4.12)
By solving equation (4.12) we can write as follows:
(4.13)
Where is pole pitch and is ratio between pole arc to pole pitch.
The divergence of magnetization vector can be calculated by using the
relation given below [98].
=0 (4.14)
As divergence is zero, so magnetic scalar potential in air region and PMs
using equations (4.4) and (4.5) is described by Laplacian equation as follows
[98].
(4.15)
The magnetic scalar potential is related with magnetic field intensity
components as follows:
(4.16)
Where is the circumferential component of magnetic field intensity
and is the axial component of the magnetic field intensity.
in the air spaces
in the magnet region
52
4.2.4 2-D Analytical Method
For the computation of magnetic field with the analytical method, the mean radius
approach that is presented in [100-102] and the rectangular coordinate system is used.
Therefore, the 3-D geometry of the AFPMSG is converted into a 2-D linear model in
which X-axis and Y-axis represent circumferential and axial directions respectively.
Computation of no load flux density components by a 2-D analytical method in the air
gap and magnet regions is derived from Maxwell's equations solution by applying
boundary conditions.
Figure 4.3 shows the AFPMSG linear model for computation of no load magnetic
field due to the lower rotor permanent magnets.
Figure 4.3 Linear representation of the AFPMSG for the lower rotor.
For permanent magnet machines with linear demagnetization characteristics, the
Laplacian equation governs the scalar magnetic potential in both air as well as
permanent magnet regions [102]. The general solutions of Laplacian equation in air
and magnet regions are given by Equations (4.17) and (4.18), respectively.
1 2
1,3,5,...
( )cos( )n y n y
p p
p
n xI III
n
D e D e
(4.17)
3 4
1,3,5,...
( )cos( )n y n y
p p
p
n xII IV
n
D e D e
(4.18)
Where φ is the magnetic scalar potential with subscripts represents
corresponding regions, y is the axial height, τp is pole pitch, n is harmonic order, x
is the circumferential distance and D1 to D4 are the unknown coefficients to be
determined by applying boundary conditions.
x
y hm NRegion II
(lower magnets)
Region I
(air space)
τm
S
Back iron
Back iron
L
53
The coefficients D1 to D4 in the above expressions are determined by imposing the
boundary conditions. The PMs magnetic field must satisfy boundary conditions given
in Equation (4.19)-(4.21) [99].
0 0( , ) ( , ) 0
( , ) ( , ) 0
xI xIIIy y
xII xIVy L y L
H x y H x y
H x y H x y
(4.19)
( , ) ( , )
( , ) ( , )
m m
m m
yI yIIy L h y L h
xI xIIy L h y L h
B x y B x y
H x y H x y
(4.20)
( , ) ( , )
( , ) ( , )
m m
m m
yIII yIVy L h y L h
xIII xIVy L h y L h
B x y B x y
H x y H x y
(4.21)
Where Hx is magnetic field intensity circumferential components and By is
magnetic flux density axial component.
By employing the above mentioned boundary conditions, we get the values of the
coefficients given by Equations (4.22)-(4.25)
1
sinh
2
m pn pn hM
Dn
(4.22)
2
sinh
2
m pn pn hM
Dn
(4.23)
3
sinh ( )
2 p
m pn p
n L
n L hMD
n e
(4.24)
4
sinh ( )
2 p
m pn p
n L
n L hMD
n e
(4.25)
Here,
( ) ( )cosh( )sinh( ) cosh( )sinh( )m pm m m
p p p p
n L hn h n L h n h
r
and,
54
0
sin( 2)2
2
prn p
p
nBM
n
Where hm is axial length of the magnet, αp is pole arc to pole pitch ratio, L is axial
height of machine, Br is residual flux density of PM, µo is permeability of free space
and µr is relative permeability.
By substituting the above computed coefficients into Equations (4.17) and (4.18)
and by solving for the magnetic field, we get the circumferential and axial
components of magnetic field. Magnetic field components due to lower magnets in air
gap and magnet regions are given by Equations (4.26) –(4.29) .
0 0
0
1,3,5,...
sinsinh sin
IxI xI
m p
n p p
n
B Hx
n hM n y n x
(4.26)
0 0
0
1,3,5,...
sincosh cos
IyI yI
m p
n p p
n
B Hy
n hM n y n x
(4.27)
0
0
1,3,5,...
sin ( )sinh ( ) sin
IIxII xII x
m p
n p p
n
B H Mx
n L hM n L y n x
(4.28)
0 0 0
1,3,5,...
1
sin ( )cosh ( ) cos
IIyII yII y y n
n
r m p
p p
B H M M My
n L hn L y n x
(4.29)
Here,
0 r
55
Where Bx is circumferential component of flux density, By is axial component of
flux density, Hx is circumferential component of field intensity and Hy is axial
component of field intensity.
Figure 4.4 shows the linear model of the AFPMSG for the computation of no load
magnetic field due to upper rotor magnets. Circumferential and axial components of
magnetic flux density due to upper rotor magnets in air gap and magnet regions are
given by Equations (4.30) -(4.33) .
0 0 0
1,3,5,...
sin
sinh ( ) sin
m pIIIxIII xIII n
n
p p
n hB H M
x
n L y n x
(4.30)
Figure 4.4 Linear representation of the AFPMSG for the upper rotor.
0 0 0
1,3,5,...
sin
cosh ( ) cos
m pIIIyIII yIII n
n
p p
n hB H M
y
n L y n x
(4.31)
0 0
1,3,5,...
sin ( )
sinh sin
m pIVxIV xIV x n
n
p p
n L hB H M M
x
n y n x
(4.32)
0 0 0
1,3,5,...
1
sinh ( )cosh cos
IVyIV yIV y y n
n
r m p
p p
B H M M My
n L hn y n x
(4.33)
x
y
hm N
Region III
(air space)
Region IV
(upper magnets)
τm
S
Back iron
Back iron
L
56
The armature reaction refers to magnetic field produced by currents in stator coils
and their interaction with field flux. Here disc armature winding is considered as thin
current sheets. Figure 4.5 shows linear model of AFPMSG for computation of
armature reaction field. The axial component of armature reaction is given by
Equation (4.34) [104].
0
cosh 2cosh ( ) cos
sinhy n
npL RB K np L y R npx R
npL R
(4.34)
For computing the Beff, the effect of both radius R and axial position y are
considered. These variables are considered because for a specific axial position y air
gap flux density is varying with radial position R. The effective no load flux density
Beff is given by Equation (4.35) .
Figure 4.5 Linear representation of the AFPMSG coil region by current sheet.
2 2
1 12 1 2 1
1( , )
( )( )
R y
eff wR y
B K B R y dRdyR R y y
(4.35)
Here, B(R,y) is the sum of magnetic field, Kw is the winding factor, R1 and R2 are
inner and outer radii of rotors and y1 and y2 are axial positions of lower and upper
surfaces of the coil region.
The back EMF is computed by considering axial and circumferential components
of twin rotor magnets. The back EMF Eb in the air gap is given by Equation (4.36) .
2 1
2 26 ( )ph
b eff
fTE R R B
p (4.36)
Where Tph the number of turns per phase.
U-UU
x
y
Xc
Back iron
Back iron
L
57
4.2.5 Characteristics Analysis
Figure 4.6(a) and Figure 4.6(b), shows magnetic flux density axial and
circumferential components due to lower and upper magnets for the air gap region
respectively. It can be seen that both magnetic flux density components due to lower
magnet decrease as y increases up to coil's center, i.e., 19 mm. Furthermore, results
show that magnetic flux density components due to upper magnets increase as y
increases from coil's center, i.e., 19 mm.
Figure 4.7(a) and Figure 4.7(b), shows magnetic flux density axial and
circumferential components due to lower and upper magnets for the magnet region
respectively. The results show that both magnetic flux density components due to the
lower magnet increase as y increase up to the magnet surface. The results also show
that both magnetic flux density axial and circumferential components due to the upper
magnet decrease as y increases from magnet surface.
Figure 4.6 (a) Magnetic field's axial component of air region (b) Magnetic field's
circumferential component of air region.
0 10 20 30 40
-0.50
-0.25
0.00
0.25
0.50
By (T
)
Distance (mm)
Due to lower PMs
y=13 mm
y=19 mm
y=25 mm
Due to upper PMs
y=13 mm
y=19 mm
y=25mm
0 10 20 30 40
0.00
0.25
0.50 Due to lower PMs
y=13 mm
y=19 mm
y=25 mm
Due to upper PMs
y= 13 mm
y= 19 mm
y= 25 mm
Bx (T
)
Distance (mm)
(a)
(b)
58
Figure 4.7 (a) Magnetic field's axial component of magnet regions. (b) Magnetic
field's circumferential component of magnet regions.
The axial component of the resultant armature reaction at the mean axial position
(air gap region) at the rated current of 7 Arms is shown in Figure 4.8, by using
Equation (4.34) . Magnetic field due to armature reaction is highest at phase band
edges. In addition, the result shows that the resultant field due to armature reaction is
minute compared to no load magnetic field, and hence can be ignored. The
computation of resultant magnetic field is done by adding magnetic field's axial and
circumferential components due to both rotor discs PMs. Figure 4.9 shows resultant
magnetic field. It is computed at mean radius and axial height.
The back EMF is computed by using Equation (4.36) . The calculated back EMF
is at the mean radius and axial height. The computational time for the back EMF
using the analytical method is less than one minute, whereas as it is around 15 hours
using 3-D FEA. Thus, the analytical method shows rapid characteristics analysis. The
back EMFs computed with 2-D analytical method as well as with 3-D FEA are shown
on Figure 4.10. The back EMF computed by using analytical method and 3-D FEA is
0 25 50 75
-0.8
-0.4
0.0
0.4
0.8
Due to lower PMs
y=2 mm
y=6.0 mm
y=10.0 mm
Due to upper PMs
y=28 mm
y=32 mm
y=36 mm
By (T
)
Distance (mm)
10 20 30 40
0.00
0.25
0.50
0.75
Due to lower PMs
y=2.0 mm
y=6.0 mm
y=10.0 mm
Due to upper PMs
y=28 mm
y=32 mm
y=36 mm
Bx (T
)
Distance (mm)
(a)
(b)
59
almost equal. The back EMF fundamental harmonic component is 92% and 90.6%
using the analytical method and 3-D FEA, respectively. Table 4.2 shows the summary
of results acquired with 2-D analytical model and 3-D FEA. The VTHD is higher for
the 3-D FEA analysis because it considers the nonlinear field characteristics.
Figure 4.8 Armature reaction field.
Figure 4.9 Resultant magnetic field.
Figure 4.10 Back EMF waveforms comparison using 2-D analytical method and 3-D
FEA of the initial model.
20 30 40 50
0.00
0.01
By (T
)
Distance (mm)
Armature Reaction Field
y=19 mm
0 75 150 225 300
-0.50
-0.25
0.00
0.25
0.50
Bef
f [T
]
Electrical Angle [deg.]
Resultant Flux Density
0 75 150 225 300
-50
-25
0
25
50
Eb [
V]
Electrical Angle [deg.]
Inital Model
2-D Analytical Method
3-D FEM
60
Table 4.2 Initial model performance comparison using 2-D analytical and 3-D FEA
Parameters Units 2-D Analytical Method 3-D FEA
Back EMF Vpeak 65.7 65.3
VTHD % 2.5 3.15
4.3 Optimization of the AFPMSG using 2-D Analytical Method
The VTHD varies with the d and hm, as shown in Figure 4.11. The VTHD is
calculated by computing the back EMF using the 2-D analytical method. It is clear
from Figure 4.11 that the VTHD increases rapidly as the interpolar separation
between the magnet increases. The VTHD also varies with the height of the magnet,
but this variation is significantly smaller in comparison.
The optimization of the VTHD % is made with the design variable, interpolar
separation, d, and axial height of magnets, hm, while maintaining the back EMF > 65
Vpeak. Figure 4.12 shows the selected design variable and their optimal values. For the
initial model under consideration, hm is equal to 10.0 mm and d is equal to 13.23 mm.
An optimal design process employing a developed 2-D analytical method is shown in
Figure 4.13. The genetic algorithm (GA) and direct search methods are used to find
design variables and objective functions optimized values.
Figure 4.11 VTHD trend.
The VTHD has a value of 2.5% for the initial model by 2-D analytical method. At
the optimized values of d and hm provided by the GA and direct search method, the
VTHD reduces to 0.39%. The VTHD % is also computed using 3-D FEA. The result
shows that a considerable reduction in the VTHD is obtained as the result of
optimization using the analytical method in the optimized model. The VTHD has a
6
5
4
3
2
1
0
VT
HD
(%
)
1110
98
76
hm (mm)
6 78
9 1011
1213 14
15
d (mm)
61
value of 3.15% and 1.5 % of initial and optimized models using FEA. The percentage
decrease achieved in VTHD is 52.38% as the result of the optimization using
analytical method. Since the 3-D FEA considers the nonlinear characteristics, the
VTHD is slightly and consistently higher than the analytical method results.
Figure 4.12 Selected design variables and their optimal values.
Figure 4.13 Optimal design process.
A comparison of back EMF of the optimized model, using 2-D analytical method,
and 3-D FEA, is presented in Figure 4.14. It is evident that the back EMF of both the
analytical and 3-D FEA is consistent. An analysis of back EMF waveforms is
performed to determine the VTHD and fundamental harmonic component. Figure
4.15 shows an initial as well as optimized models harmonic components comparison
using 3-D FEA. Since the considered AFPMSG is Y-connected, only the comparison
of belt harmonics is considered in this paper. The result shows that the optimized
Coil
y hm N
τp
S
Back Iron
Back Iron
L
N S
d
Objective function
Minimize VTHD
Constraint
Eb > 65 Vpeak
Design variables
6mm < d< 15 mm
6 mm < hm < 11 mm
Optimized variables
d - 6.76 mm
hm - 8.8 mm
Initial variables
d - 13.23 mm
hm - 10 mm
x
Satisfy the
Target ?
End
Yes
No
Adjust the
design variables
Start
Determine the objective functions
and design variables
2-D analytical method
Search the optimal value using
genetic and direct search algorithm
62
model has an increased fundamental harmonic component. The result also shows the
reduction in belt harmonics components. Performance comparison of the optimized
model with 2-D analytical method as well as 3-D FEA is presented in Table 4.3. The
result shows that the back EMF of the optimized model using 2-D analytical method
as well as 3-D FEA are almost same. The back EMF fundamental harmonic
component is 93% and 92% using the analytical method and 3-D FEA,
correspondingly. Finally, a comparison of initial design and the optimized design of
the AFPMSG is tabulated in
Table 4.4. The back EMF of the initial and optimized model is almost the same.
The percentage decrease in the VTHD is 52.38 % as a result of the optimization. In
addition, the optimized model is more compact in comparison to the initial model.
Figure 4.14 Optimized model back EMF comparison with 2-D analytical method and
3-D FEA.
Figure 4.15 Belt Harmonics comparison.
0 75 150 225 300
-75
-50
-25
0
25
50
Eb [
V]
Electrical Angle [deg.]
Optimized Model
2-D Analytical Method
3-D FEM
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Am
pli
tud
e o
f th
e c
om
po
ne
nts
Harmonic order n
Initial Model
Optimized Model
0.008
0.004
0
5 7
63
Table 4.3 Optimized model performance comparison with 2-D analytical method and
3-D FEA
Parameters Units 2-D analytical method 3-D FEA
Back EMF Vpeak 65.5 65.4
VTHD % 0.39 1.5
Figure 4.16 Flux density distribution plots by 3D-FEA: (a) Initial model (b)
Optimized model.
In order to obtain a load analysis of the AFPMSG's initial and optimized models, a
load resistor of 6.8 ohms is connected across each phase. Figure 4.16(a) and Figure
4.16(b), shows initial and optimized models flux density distributions under load
condition. The maximum flux density (Bmax) is almost 1.8 T and 2.0 T for the initial
and optimized models respectively, which occurs at back iron. The optimized model
increased flux density is because of its increased magnet surface area caused by
decreased interpolar separation as can be seen from the Table 4.4.
An output torque comparison of initial and optimized models is shown in Figure
4.17. An increase in output torque of optimized model is achieved, as compared with
initial model. Output torque of initial model is 6.9 Nm and that of the optimized
model is 7.74 Nm. Furthermore, a reduction in optimized model torque ripple is
achieved. The initial and optimized models torque ripple are 45% and 36% by 3D-
FEA.
2.4
1.8
1.2
0.6
0.0
Magnetic Flux Density
Contour Plot T2.4
1.8
1.2
0.6
0.0
Magnetic Flux Density
Contour Plot T
(a) (b)
64
Table 4.4 Initial and optimized model comparison
Parameters Units Initial Model Optimized Model
Interpolar separation mm 13.23 6.76714
Height of magnet mm 10.0 8.81239
Axial height of machine mm 46 43.62
Back EMF V 65.3 65.4
VTHD % 3.15 1.5
Torque Nm 6.9 7.74
Torque ripple (Tpk2pk) % 45 36
Figure 4.17 Torque comparison of the initial and optimized model by 3D- FEA.
4.4 Summary
A 2-D analytical method to compute back EMF of coreless dual rotor
AFPMSG by solving Maxwell’s equations is presented in this chapter. The 2-D
analytical method results are verified with 3-D FEA. Furthermore, the VTHD % is
reduced through optimization with the developed 2-D analytical method. The VTHD
of initial and optimized models is compared using 3-D FEA and results show that the
VTHD is 1.5%, which is a considerable improvement over the previous 3.15%.
Furthermore, the optimal design exhibits reduced torque ripple with higher average
output torque, as compare to the initial model. The time saved due the 2-D analytical
method proves the advantages of the analytical technique against the time-consuming
3-D FEA method. Therefore, the developed 2-D analytical method aids the design of
the AFPMSG due to its reduced time over the FEA.
0.0000 0.0025 0.0050 0.0075 0.0100 0.0125
0
-2
-4
-6
-8
-10
To
rqu
e (N
m)
Time (sec)
Optimized Model Initial Model
Tavg = 6.9
Tripple = 45%
Tavg = 7.74
Tripple = 36%
65
Chapter 5 Reduction of Torque Ripple in an AFPMSG
using Arc Shaped Trapezoidal Magnets in an
Asymmetric Overhang Configuration
66
5.1 Introduction
For the smooth working of the coreless AFPM machine, eradication of the torque
ripple is necessary. However, similar to the other types of AFPMSG, coreless
AFPMSG also produce torque ripples. The major sources of torque ripple in coreless
AFPMSG are grouped into the following categories: nonsinusoidal back EMF,
cogging torque and saturation of the magnetic circuit [78, 105]. A lot of methods,
including magnet shaping, pole arc to pole pitch ratio, coil shapes and winding
configuration, have been proposed to eradicate torque ripples in coreless AFPM
machines [70, 103, 106-108].
The improved performance of the machines requires leakage flux minimization
along with increasing air gap flux. For RFPM machine, increase in air gap flux along
with minimization of leakage flux have been proposed with PM in an overhang
configuration [86, 109]. Overhang techniques, including optimizing the rotor
overhang variation and PM overhang in the tangential direction, have been proposed
for enhancing performance of AFPM machines [110, 111].
In this chapter, a topology of coreless AFPMSG with arc-shaped trapezoidal PM’s
is presented to reduce torque ripples. However, proposed model output torque is
reduced compared to the AFPMSG's conventional model. The reduction in proposed
model output torque is because of effective air gap increase. Therefore, proposed
model is optimized with PM in an asymmetric overhang configuration. The
experiments were designed by using LHS for design variables. The objective
functions and constraints are approximated using Kriging method. Finally GA is
utilized to obtain optimal results. 3D FEA is utilized for magnetic field analysis due to
tridimensional electromagnetic nature of AFPMSG. The rest of chapter is arranged in
this manner: Section 5.2 presents the conventional and proposed models of coreless
AFPMSGs. This is followed by Section 5.3, which contains an optimization process
for the proposed model and its results. In Section 5.4, a conclusion of the overall
research work is presented.
5.2 Comparison between the proposed and conventional Model
In this section, the design process of the coreless AFPMSG, proposed magnet
shape and a comparative analysis of the conventional and proposed basic models is
67
presented. AFPMSG with a flat trapezoidal magnet is called the conventional model
and with a proposed arc-shaped trapezoidal magnet is called the proposed model.
5.2.1 Proposed Magnet Shape
In RFPM machines, magnet length is along stack length, where the circumference
includes both sides of the stack and which have the same length. Therefore, using a
rectangular magnet shape will result in effective utilization of the rotor surface area.
However, for the AFPMSG, the length of the magnet is from the inner to outer rotor
back iron diameter. Rotor back iron outer circumference is greater than the inner
circumference. Therefore, in order to increase effective utilization of the rotor surface
area, a trapezoidal shape is more advantageous than a circular or rectangular shape
[67].
Furthermore, using an arc-shaped PM in an RFPM machine reduces torque ripple
and cogging torque compared to the flat PM because the arc-shaped PM makes air
gap flux more sinusoidally distributed and increases effective air gap length. Air gap
length in arc-shaped PM is not the same over one pole; specifically, it is at a
minimum in the middle of the magnet and at a maximum at the edges of the magnets.
The rise in effective air gap length decreases air gap flux, which in turns reduces the
overall cogging torque [109].
Similarly, in the AFPMSG, an arc-shaped PM will reduce torque ripple and
cogging torque compared to flat PM. Furthermore, this arc-shaped PM will result in
an unsymmetrical air gap as can be seen from Figure 5.1. This unsymmetrical air gap
reduces the air gap flux in the AFPMSG. The decrease in the air gap flux will result
into the reduced in the cogging torque and is expressed as follows.
(4.1)
Where φg is the flux in air gap, R is air gap reluctance and θ is the rotor position.
Figure 5.1 PM shapes: (a) Conventional magnet (b) Proposed magnet
(a) (b)
68
Figure 5.2 shows the flat trapezoidal and proposed arc-shaped trapezoidal PMs on
the rotor back iron. The variables Wo, Wi, Hie, Hoe and Lm represent the PM outer
width, PM inner width, PM inner edge height, PM outer edge height and PM length,
respectively. The impact on the performance of the AFPMSG with the proposed
shape PM is discussed in Section 5.2.3.
Figure 5.2 Parameters of the PM shapes: (a) trapezoidal (b) arc-shaped trapezoidal
Table 5.1 Conventional and proposed models parameters
Parameters Conventional
Model
Proposed
Model Parameters
Conventional
Model
Proposed
Model
Speed 1100 rpm Nph 396
Poles 12 Do/Di 152/84.6 mm
Coils 9 Magnet
volume 6988 mm3
Air gap 1.5 mm Coil
resistance 0.23 Ohms
Yoke height 5.5 mm Hie 10 mm 8.95 mm
Br 1.2 T Hoe 10 mm 7 mm
Lm 30 mm Wi 16.8 mm 18.3 mm
Coil height 15 mm Wo 28.6 mm 29.2 mm
5.2.2 Design Process
The topology selected for the design of AFPMSG consists of a single coreless
stator and dual rotor back iron having surface mounted magnets. A 1 kW coreless
AFPMSG is designed by using [15]. A design process flow chart is shown in Figure
5.3, while conventional model is based on [12]. The computed performance
Hie
Wi
Lm
WoHoe
(a) (b)
69
parameters obtained using the sizing equation are verified by 3-D FEM. Various
parameters of coreless AFPMSG conventional and proposed model are presented in
Table 5.1. The variables Br, Nph, Do and Di represent the PM residual flux density,
turns per phase, rotor outer diameter and rotor inner diameter, respectively. The
height of a conventional PM is 10 mm throughout. However, the height of the
proposed PM is not constant throughout. The height of the proposed PM is 10 mm in
the middle, 8.95 mm at inner edge and 7 mm at outer edge. The proposed model air
gap length is not same over one pole. Its maximum is at the outer edges and its
minimum is at middle of the magnet. The air gap length is 2.5 mm at inner edges and
4.5 mm at outer edges. Thus, the air gap length in the proposed model varies between
1.5 mm and 4.5 mm. However, the air gap length in the conventional model is 1.5 mm
throughout. The length and volume of the proposed and conventional shape PM are
kept constant. The steel sheet used for the rotor back iron is 50JN1000. Performance
analysis of the conventional and proposed AFPMSG models is presented in the
following section.
Figure 5.3 Flow chart of the design process.
Power, voltage, frequency, speed, efficiency
Magnetic & electric loading, voltage form factor,
winding factor, number of coils, rotor diameter ratio
Flux per pole, turns/phase, voltage, current, power
Poles, outer and inner diameter of rotor yoke, coil &
pole pitch
Conductor area, magnet, yoke, coil, total height of
machine
Verified by 3-D FEM
Satisfy the target?
End
Yes
No
70
Figure 5.4 Exploded AFPMSGs with concentrated windings: (a) conventional model
(b) proposed model.
5.2.3 AFPMSG Conventional and Proposed Models
Performance Comparison
In order to achieve an accurate characteristic analysis, 3D-FEA is used in this
thesis, specifically JMAG designer ver. 14.1 is used as the 3D-FEA tool. The volume
of the magnet is kept constant for the performance comparison between the
conventional and proposed models. In order to maintain the same volume for the
conventional trapezoidal and arc-shaped trapezoidal PMs, proposed model's pole arc
to pole pitch ratio is adjusted as shown in Table 5.1. Figure 5.4 presents an exploded
view of the AFPMSG's conventional and proposed models. The coils of various
phases are arranged in the circumferential direction.
The conventional model flux density distributions of entire model and its coil
region are illustrated in Figure 5.5(a) and Figure 5.5(b), respectively. The flux density
distributions of proposed model as well as its coil region are presented in Figure
5.6(a) and Figure 5.6(b), respectively. The maximum flux density (Bmax) for both
models is almost 1.8 T, which occurs at the rotor back iron. Bmax in the coil region is
0.6 T for the conventional and proposed models. The effect of an increase in air gap
length can also be observed in coil region flux density distribution plots. It is evident
from flux density plot of coil regions of both models, the proposed model overall flux
density is lower. This decrease in flux density is because of increased overall effective
air gap length.
Coil phase a
Coil phase b
Coil phase c
Rotor back iron
PM north pole
PM south pole
Semi- spherical trapezoidal PM
Trapezoidal PM
(a) (b)
71
Figure 5.5 Flux density distribution: (a) entire conventional model (b) coil region.
Figure 5.6 Flux density distribution: (a) entire proposed model (b) coil region.
Figure 5.7 Back EMF waveforms for the conventional and proposed models.
(a) (b)
1.80
1.35
0.90
0.45
0.0
Magnetic Flux DensityContour Plot T
0.60
1.45
0.30
0.15
0.0
Magnetic Flux DensityContour Plot T
(a) (b)
1.80
1.35
0.90
0.45
0.0
Magnetic Flux DensityContour Plot T
0.60
1.45
0.30
0.15
0.0
Magnetic Flux DensityContour Plot T
0 2 4 6 8 10 12
-75
-50
-25
0
25
50
75
Bac
k E
MF
[V]
Time [msec]
Conventional model Phase A Phase B Phase C
Proposed model Phase A Phase B Phase C
Vrms = 51.1 Vrms = 47.4
72
The back EMF of conventional and proposed models is shown in Figure 5.7. There
was an overall 3.7 V decrease observed in the magnitude of the back EMF with the
proposed model. The decrease in the magnitude of the back EMF is due to the
increased effective air gap. Fundamental harmonic component of conventional model
is 92.6% and that of the proposed model is 94.8%. The THD of the conventional
model is 1.9% and that of the proposed model is 1.4%. A 26.3% reduction in THD is
achieved with the proposed model. The increase in the fundamental harmonic
component and reduction in the THD is because of more sinusoidal flux density
distribution of proposed model.
A comparison of cogging torque of conventional as well as proposed models is
presented in Figure 5.8. The proposed model cogging torque is considerably reduced
to that of conventional model. Decrease in peak-to-peak cogging torque is 1.0 Nm.
The proposed model has a 71.4% reduction in cogging torque when compared with
conventional model. The decrease in cogging torque of proposed model is due to the
increased effective air gap and arc-shaped trapezoidal PM.
In order to obtain an output torque, a load resistor of 6.8 ohms is connected across
each phase. The average torque comparison of the both models is presented in Figure
5.9. With proposed model the decrease in average torque is 1.2 Nm. The average
torque of proposed model is reduced because of increase in effective air gap length.
The proposed model has a 64.52% reduction in the torque ripple when compared with
conventional model. The reduction in torque ripple is achieved due to the decrease in
the cogging torque.
Figure 5.8 Cogging torque comparison of the conventional and proposed models.
0 2 4 6 8 10 12
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
Co
gg
ing
To
rqu
e [
Nm
]
Time [msec]
Conventional Model Proposed Model
Tpk2pk = 1.4
Tpk2pk = 0.4
73
Figure 5.9 Torque comparison of conventional and proposed models.
Table 5.2 shows a performance comparison between conventional and proposed
models. Proposed model has reduced THD, cogging torque and torque ripples due to
the arc-shaped trapezoidal PM and increased air gap. However, the Vrms and the
output torque of the proposed model have decreased compared to that of the
conventional model. The decrease in Vrms and output torque is due to increased
effective air gap. Optimization of proposed model will be performed to make the
output torque more competitive compared with the conventional model.
Table 5.2 Performance comparison between the conventional and proposed models of
coreless AFPMSGs
Parameter Unit Conventional
model
Proposed
model
Back EMF Vrms 51.1 47.4
Back EMF fundamental harmonic % 92.6 94.8
THD % 1.9 1.4
Cogging Torque Tpk2pk Nm 1.40 0.4
Torque Tavg Nm 9.1 7.9
Torque Ripples % 15.5 5.5
5.3 Proposed Model Optimization
To achieve an increased output torque compared to the conventional model,
optimization of the proposed model is performed in this section. The volume of PM is
kept constant throughout optimization process.
0 2 4 6 8 10 12
0
-2
-4
-6
-8
-10
To
rqu
e [
Nm
]
Time [msec]
Conventional Model Proposed Model
Tavg = 7.9
Tripple = 5.5 %Tavg = 9.1
Tripple = 15.5 %
74
5.3.1 Selection of Design Variables
In this section, in order to develop an optimized model, an asymmetric magnet
overhang is used. The PM length along both inner as well as outer radii is varied. An
asymmetric magnetic overhang is when the length of PM is unequally extended over
rotor disc's inner and outer radii. PM overhang is used for increasing coil flux linkage.
The length of the PM is varied from outer radius of disc to outer radius of coils and
from inner radius of disc to inner radius of coils. By varying the PMs dimension
between these limits the overall structure will remain same. The PM overhang is used
to increase output torque due to increased coil's flux linkage.
The design variables X1 and X2 are the inner and outer overhang lengths of the
magnets to alter the generator's performance as shown in Figure 5.10(a). The pole arc
to pole pitch ratio X3 (as given in Figure 5.10(b)) and height of PM arc X4 (as given in
Figure 5.10(c)) are also taken as design variables. The asymmetric PM overhang is
shown in Figure 5.10(a). Various combinations of these design variables are obtained
via LHS. The volume of the arc-shaped trapezoidal magnet is kept fixed for each of
the combinations by adjusting the trapezoid height (Ht).
Figure 5.10 Design variables: (a) Asymmetric PM overhang, (b) Top view (c) Cross-
sectional view.
Objective functions
Minimize VTHD and Cogging torque
Design variables
0 < X1 > 8
0 < X2 > 14
0.45 < X3 > 0.8
1 < X4 > 4
Constraint
Back EMF >=51 Vrms
(a)
X3= Pole width
Pole pitch
N S
NS
N
Coil phase a
Magnet
Inner overhang (X1) Outer overhang (X2)
Magnet
Rotor Back Iron
(c)
(b)
X4
75
Figure 5.11 Arc-shaped trapezoidal PM parameters.
Figure 5.11, shows the arc-shaped trapezoidal PM for the calculation of magnet
volume. The variables R, θoa, θia, Hoa, and Hir represent the radius of the arc, angle of
the outer arc, angle of the inner arc, outer arc height and inner rectangle height,
respectively. The volume of the arc segment (Vas) is given by the following equation:
(4.2)
Where Ro along with Ri, are PMs circular array outer and inner radii, and Aia and
Aoa denote the area of inner and outer arc segment, respectively. The area of inner arc
segment is equal to sum of inner rectangle and inner arc segment areas. The Aia is
computed as follows:
(4.3)
The area of the outer arc can also be computed similarly. The total volume of the
magnet is equal to the sum of the volume of segments and the volume of the
trapezoids. The volume of a trapezoid-shaped PM (Vt) is given by the following
equation:
(4.4)
5.3.2 Optimization Process
Figure 5.12, presents an optimal design process. Initially, objective function and
design variables are chosen. The LHS method is used to design the experiments.
Taking into account the number of design variables, total number of designed
experiments is 15. The volume of the magnet is kept fixed in all design experiments to
ensure constant machine weight and hence optimal performance. Then, 3D-FEA is
used for the performance analysis. After that, Kriging method is utilized for objective
function estimation. Then, GA is used to get the optimized values of design variables
Inner side of PM
θoa
Hir
Wo
Hoa = X4
Outer side of PM
R
Wi
θia
Hie
Hoe
Hia
Ht
76
and objective functions. Finally, a 3D-FEA is performed to verify the optimal results
obtained using the design process.
Figure 5.12 Optimal design process.
Figure 5.13 Optimized model flux density distribution.
5.3.3 Optimal Design Results
Figure 5.13(a) and Figure 5.13(b) show the flux density distribution of the
optimized model and its coil region, respectively. The overhang of the PMs and the
optimized model structure can also be clearly observed from this figure. Maximum
flux density is around 1.78 T in rotor back iron and 0.52 T in the coil region of the
optimized model. The maximum flux density in rotor back iron of optimized model is
smaller than the proposed and conventional models. The maximum flux density in
Satisfy the Target ?
End
Yes
No
Adjust the design variables
Start
Determine the objective functions
and design variables
Approximate the model by Kriging
analysis
Design of experiment
(Latin hyper cube sampling)
Search the optimal value using
Genetic algorithm
3D – FEA performance analysis
Adjustment of trapezoid height for
constant volume
(a) (b)
1.78
1.335
0.89
0.445
0.0
Magnetic Flux DensityContour Plot T
0.52
1.39
0.26
0.13
0.0
Magnetic Flux DensityContour Plot T
77
coil region is lower for optimized model than for proposed and conventional models.
However, the overall flux density and flux linkage are increased in its coil region due
to PM overhang. This increase in the flux linkage of the coil will increase back EMF
and hence output torque.
Figure 5.14 Optimized model back EMF.
Figure 5.15 Cogging torque comparison of the proposed and optimized models.
The back EMF waveform of the optimized model is shown in Figure 5.14. The
increase in back EMF of the optimized 3.8 Vrms is compared to that of the proposed
model. An 8% increase in the Vrms is achieved as a result of the optimization.
A comparison of cogging torque of proposed and optimized models is shown in
Figure 5.15. The cogging torque of optimized model is also reduced compared to that
of the proposed model. The percentage decrease in peak-to-peak cogging torque is
10%.
0 2 4 6 8 10 12
-75
-50
-25
0
25
50
75
bac
k E
MF
[V
]
Time [msec]
Phase A Phase B Phase C
Vrms = 51.2
0 2 4 6 8 10 12
-0.3
-0.2
-0.1
0.0
0.1
0.2
Co
gg
ing
to
rqu
e [
Nm
]
Time [msec]
Proposed Model Optimized Model
Tpk2pk = 0.36 Tpk2pk = 0.4
78
Figure 5.16 Torque comparison of the proposed and optimized models.
Figure 5.17 Output power comparison of the proposed and optimized models.
An output torque comparison of both models is presented in Figure 5.16. Sufficient
improvement in output torque is achieved as a result of optimization. The torque
ripple of optimized model is also decreased. Furthermore, average torque of
optimized model is increased by 18.35%. A torque ripples reduction is 26.34%
compared to proposed model. In addition, output power of the optimized model is
18.23% greater than proposed model as presented in Figure 5.17.
A comparison of the design parameters is presented in Table 5.3. The volume of
the magnet is the same in both the proposed and optimized proposed models. The
total axial height of the optimized proposed model is 45.5 mm as compared to 49 mm
for the proposed model. Therefore, optimized model is more compact than proposed
model. For optimized model, values for the magnet’s inner and outer radii are 37.1
mm and 81.4 mm, compared to 43.5 mm and 74.2 mm for the proposed model,
0 2 4 6 8 10 12
0
-2
-4
-6
-8
-10
To
rqu
e [
Nm
]
Time [msec]
Proposed Model Optimized Model
Tavg = 7.9
Tripple = 5.5 %
Tavg = 9.35
Tripple = 4.06 %
0 2 4 6 8 10 12
0
200
400
600
800
1000
Ou
tpu
t P
ow
er
[W]
Time [msec]
Proposed Model Optimized Model
Pavg = 854.5 Pavg = 1010.3
79
respectively. Although the outer diameter of the PM array is increased, it is still lower
than the coil's outer diameter and hence the frame size of the machine will remain the
same.
A performance comparison of the various parameters is presented in Table 5.4. The
decrease in optimized model iron loss is 10 W compared to the proposed model. The
increase in optimized model copper loss is 15.83 W. The optimized model copper loss
increased due to increase in current caused by increase in terminal voltage. However,
the percentage increase in the efficiency of the optimized proposed model is 1.23%
compared to the proposed model.
Table 5.3 Comparison of design parameter
Parameter Units Proposed model Optimized model
X1 mm N/A 4.13
X2 mm N/A 0.9
X3 mm 0.8 0.75
X4 mm 3 1.21
Hie mm 8.95 8
Hia mm 1.05 0.26
Hoe mm 7 7.05
Total machine height mm 49 45.5
PM volume mm3 6988 6988
Table 5.4 Comparison of performance parameters
Parameter Units Proposed model Optimized model
back EMF Vrms 47.4 51.2
Output Power W 854.5 1010.3
Current Arms 6.497 7.06
Copper Losses W 87.37 103.2
Iron Losses W 17 7
Efficiency % 89.1 90.2
Torque Ripples % 5.512 4.06
Average Torque Nm 7.9 9.35
Cogging Torque Tpk2pk Nm 0.4 0.36
80
5.4 Summary
A model of a coreless AFPMSG using an arc-shaped trapezoidal PM is proposed
and investigated in this chapter. Compared to conventional model, proposed model
has a reduced cogging torque and torque ripple at the cost of a decrease in the average
torque due to an increase in effective air gap length. The proposed model is then
optimized to increase the average torque as well as to further reduce cogging torque.
The optimal design exhibits reduced torque ripple with higher average torque
compared to the conventional and proposed models. Furthermore, the efficiency of
optimized model is also competitive with proposed model. The proposed model
cogging torque and torque ripples are considerably reduced to that of conventional
model. The proposed model has a 71.4% reduction in cogging torque and 64.52%
reduction in the torque ripple when compared with conventional model. The average
torque of optimized model is increased by 18.35%, torque ripples reduction is 26.34%
and output power of the optimized model is 18.23% greater than proposed model.
Thus the optimal design shows improved performance characteristics compared with
the conventional and proposed models.
81
Chapter 6 Conclusion and Future Work
82
The main objective of this research was to develop a model of the AFPMSG for the
wind power generation with reduce ripples and improved output torque. Axial flux
configuration is chosen due to its benefit of generating increased power density and
torque density as compared to the radial flux configuration. The configuration chosen
for the AFPMSG is of coreless stator. Furthermore, the rotor of the selected
configuration consists of the two rotor disc with surface mounted PMs on them.
Coreless type configuration of the AFPMSG has the advantages of increased
efficiency due to the removal of the stator core losses. Furthermore coreless stator
dual rotor structure has advantages of balance force of attraction between the stator
and rotor. The origin of saturation phenomenon in coreless AFPMSG may be
attributed to one, a few or all of factors such as armature reaction, air gap flux density,
high density NdFeB magnets, temperature rise or thickness of the rotor back iron.
A D3 method is discussed for the design of the axial flux machine in this thesis.
Also design procedure for the coreless AFPM machine is also discussed. Furthermore,
the torque and power equations of the dual rotor single coreless stator AFPM machine
are also derived. Various equations for the calculation of the design variables is
presented. In addition 3-D FEA is discussed along with the Maxwell equations.
For the characteristics analysis of the coreless AFPMSG a 2-D analytical method
is presented. A mathematical model is developed for the no load magnetic field and
back EMF by using the analytical method. The analytical method is developed by
solving Maxwell equation along with the boundary condition method. Fourier series
are utilized for the magnetization characteristics of the magnet. Furthermore, the
effect of varying interpolar separation and magnet height is also shown.
Moreover, the optimization of the coreless AFPMSG using an analytical method is
presented. Optimization using the analytical method reduces the optimization time to
less than a minute. It is shown that VTHD and torque ripple are reduced as the results
of the optimization. Furthermore, it is also shown the torque output is increased as the
result of the optimization. 3-D FEA is utilized for comparing results with analytical
method.
Traditionally, the flat trapezoidal shaped magnet is used most commonly in the
AFPM machines. To reduce the torque ripple of AFPM machine an arc-shaped
trapezoidal PM is proposed and investigated in this thesis. The performance
comparison of the axial flux machine with both conventional shape and proposed
83
shape magnets is presented. Utilizing the arc shape magnets in the AFPM machine
resulted in reducing cogging torque and VTHD. This reduction in VTHD and cogging
torque was because of increased effective air gap with arc shape magnet.
Compared to the conventional model, the proposed model has a reduced cogging
torque and torque ripple at the cost of a decrease in the average torque due to the
increase in effective air gap length. The proposed model is then optimized to increase
the average torque and to further reduce cogging torque. The optimal design exhibits
reduced torque ripple with higher average torque compared to the conventional and
proposed models. Furthermore, the efficiency of the optimized model is also
competitive with the proposed model. The optimal design shows improved
performance characteristics compared with the conventional and proposed models.
In future a prototype of the AFPM machine with the proposed arc shaped
trapezoidal magnets will be developed. The performance of the prototype model will
be compared with the simulation results. Performance analysis of the AFPM machines
with proposed magnet shape and various other winding configurations such as double
layer concentrated and triple layer along with triple layer wave winding will be
analyzed. In order to see the demagnetization effect of the proposed PM shape
AFPMSG thermal analysis will be performed. Furthermore, AFPMSG's design will
be presented with a reduced speed range of 300 RPM to 500 RPM. In addition, slotted
and slot-less type dual rotor AFPMSG will be analyzed with proposed PM shaped PM
and their performance will be compared with coreless type AFPMSG.
84
References
85
[1] G. M. Joselin Herbert, S. Iniyan, E. Sreevalsan, and S. Rajapandian, "A review
of wind energy technologies," Renewable and Sustainable Energy Reviews,
vol. 11, pp. 1117-1145, 8// 2007.
[2] H. Polinder, J. A. Ferreira, B. B. Jensen, A. B. Abrahamsen, K. Atallah, and R.
A. McMahon, "Trends in Wind Turbine Generator Systems," IEEE Journal of
Emerging and Selected Topics in Power Electronics, vol. 1, pp. 174-185,
2013.
[3] N. Goudarzi and W. D. Zhu, "A review on the development of wind turbine
generators across the world," International Journal of Dynamics and Control,
vol. 1, pp. 192-202, 2013.
[4] H. P. D Bang, G Shrestha, Jan Abraham Ferreira, "Review of generator
systems for direct-drive wind turbines," presented at the European Wind
Energy Conference & Exhibition, Belgium, 2008.
[5] J. A. Baroudi, V. Dinavahi, and A. M. Knight, "A review of power converter
topologies for wind generators," Renewable Energy, vol. 32, pp. 2369-2385,
11// 2007.
[6] J. F. Gieras, Permanent Magnet Motor Technology New York: Marcel
Dekker, 2002.
[7] B. K. Ramasamy, A. Palaniappan, and S. M. Yakoh, "Direct-drive low-speed
wind energy conversion system incorporating axial-type permanent magnet
generator and Z-source inverter with sensorless maximum power point
tracking controller," IET Renewable Power Generation, vol. 7, pp. 284-295,
2013.
[8] A. Hassanpour Isfahani and S. Vaez-Zadeh, "Line start permanent magnet
synchronous motors: Challenges and opportunities," Energy, vol. 34, pp.
1755-1763, 11// 2009.
[9] Z. Q. Zhu, "Recent Advances on Permanent Magnet Machines," Transactions
of China Electrotechnical Society,, vol. 27, pp. 1-11, 2012.
[10] T. A. Lipo and U. o. W.-.-M. W. P. E. R. Center, Introduction to AC Machine
Design: WisPERC, University of Wisconsin, 2011.
[11] Z. Chen, J. M. Guerrero, and F. Blaabjerg, "A Review of the State of the Art
of Power Electronics for Wind Turbines," IEEE Transactions on Power
Electronics, vol. 24, pp. 1859-1875, 2009.
[12] W. L. K. T. Chau, and C. H. T. Lee, "Challenges and opportunities of electric
machine for renewable energy," Progress In Electromagnetics Research B,
vol. 42, pp. 45–74, 2012.
[13] H. Li and Z. Chen, "Overview of different wind generator systems and their
comparisons," IET Renewable Power Generation, vol. 2, pp. 123-138, 2008.
[14] H. Polinder, "Overview of and trends in wind turbine generator systems," in
2011 IEEE Power and Energy Society General Meeting, 2011, pp. 1-8.
[15] T. Davenport, "Improvement in propelling machinery by magnetism and
electro-magnetism," ed: Google Patents, 1837.
[16] I. Gottlieb, Practical Electric Motor Handbook: Elsevier Science, 1997.
[17] L. H. Hansen, P. H. Madsen, F. Blaabjerg, H. Christensen, U. Lindhard, and
K. Eskildsen, "Generators and power electronics technology for wind
turbines," in Industrial Electronics Society, 2001. IECON'01. The 27th Annual
Conference of the IEEE, 2001, pp. 2000-2005.
[18] W. Cao, Y. Xie, and Z. Tan, "Wind turbine generator technologies," in
Advances in Wind Power, ed: InTech, 2012.
[19] T. Ackermann, Wind power in power systems: John Wiley & Sons, 2005.
86
[20] U. K. Madawala, T. Geyer, J. B. Bradshaw, and D. M. Vilathgamuwa,
"Modeling and analysis of a novel variable-speed cage induction generator,"
IEEE Transactions on Industrial Electronics, vol. 59, pp. 1020-1028, 2012.
[21] J. L. Domínguez-García, O. Gomis-Bellmunt, L. Trilla-Romero, and A.
Junyent-Ferré, "Indirect vector control of a squirrel cage induction generator
wind turbine," Computers & Mathematics with Applications, vol. 64, pp. 102-
114, 2012.
[22] L. H. Hansen, L. Helle, F. Blaabjerg, E. Ritchie, S. Munk-Nielsen, H. W.
Bindner, et al., "Conceptual survey of generators and power electronics for
wind turbines," 8755027431, 2002.
[23] N. Patil and Y. Bhosle, "A review on wind turbine generator topologies," in
Power, Energy and Control (ICPEC), 2013 International Conference on,
2013, pp. 625-629.
[24] A. Beainy, C. Maatouk, N. Moubayed, and F. Kaddah, "Comparison of
different types of generator for wind energy conversion system topologies," in
Renewable Energies for Developing Countries (REDEC), 2016 3rd
International Conference on, 2016, pp. 1-6.
[25] R. M. Kamel, "Effect of wind generation system types on Micro-Grid (MG)
fault performance during both standalone and grid connected modes," Energy
Conversion and Management, vol. 79, pp. 232-245, 2014.
[26] R. A. McMahon, X. Wan, E. Abdi-Jalebi, P. J. Tavner, P. C. Roberts, and M.
Jagiela, "The BDFM as a generator in wind turbines," in Power Electronics
and Motion Control Conference, 2006. EPE-PEMC 2006. 12th International,
2006, pp. 1859-1865.
[27] K. Protsenko and D. Xu, "Modeling and control of brushless doubly-fed
induction generators in wind energy applications," in Applied Power
Electronics Conference, APEC 2007-Twenty Second Annual IEEE, 2007, pp.
529-535.
[28] S. Shao, T. Long, E. Abdi, and R. A. McMahon, "Dynamic control of the
brushless doubly fed induction generator under unbalanced operation," IEEE
Transactions on Industrial Electronics, vol. 60, pp. 2465-2476, 2013.
[29] A. Arifin and I. Al-Bahadly, "Switched reluctance generator for variable speed
wind energy applications," Smart Grid and Renewable Energy, vol. 2, pp.
227-36, 2011.
[30] R. E. Betz and M. G. Jovanovic, "The brushless doubly fed reluctance
machine and the synchronous reluctance machine-a comparison," IEEE
Transactions on Industry Applications, vol. 36, pp. 1103-1110, 2000.
[31] M. A. Rahman and P. Zhou, "Analysis of brushless permanent magnet
synchronous motors," IEEE Transactions on Industrial Electronics, vol. 43,
pp. 256-267, 1996.
[32] A. Beainy, C. Maatouk, N. Moubayed, and F. Kaddah, "Comparison of
different types of generator for wind energy conversion system topologies," in
2016 3rd International Conference on Renewable Energies for Developing
Countries (REDEC), 2016, pp. 1-6.
[33] C. Yicheng, P. Pillay, and A. Khan, "PM wind generator topologies," IEEE
Transactions on Industry Applications, vol. 41, pp. 1619-1626, 2005.
[34] O. Gutfleisch, "Controlling the properties of high energy density permanent
magnetic materials by different processing routes," Journal of Physics D:
Applied Physics, vol. 33, p. R157, 2000.
87
[35] A. M. Pawlak, Sensors and Actuators in Mechatronics: Design and
Applications: Taylor & Francis, 2006.
[36] A. Chen, R. Nilssen, and A. Nysveen, "Performance Comparisons Among
Radial-Flux, Multistage Axial-Flux, and Three-Phase Transverse-Flux PM
Machines for Downhole Applications," IEEE Transactions on Industry
Applications, vol. 46, pp. 779-789, 2010.
[37] K. Chau and W. Li, "Overview of electric machines for electric and hybrid
vehicles," International Journal of Vehicle Design, vol. 64, pp. 46-71, 2014.
[38] T.-Y. Lee, M.-K. Seo, Y.-J. Kim, and S.-Y. Jung, "Motor Design and
Characteristics Comparison of Outer-Rotor-Type BLDC Motor and BLAC
Motor Based on Numerical Analysis," IEEE Transactions on Applied
Superconductivity, vol. 26, pp. 1-6, 2016.
[39] M. Cheng, W. Hua, J. Zhang, and W. Zhao, "Overview of Stator-Permanent
Magnet Brushless Machines," IEEE Transactions on Industrial Electronics,
vol. 58, pp. 5087-5101, 2011.
[40] P. Zheng, Q. Zhao, J. Bai, B. Yu, Z. Song, and J. Shang, "Analysis and Design
of a Transverse-Flux Dual Rotor Machine for Power-Split Hybrid Electric
Vehicle Applications," Energies, vol. 6, p. 6548, 2013.
[41] N. Tesla, "Electro-magnetic motor," ed: Google Patents, 1888.
[42] S. Khaliq, "Novel Dual Stator Axial Flux Brushless Doubly Fed Reluctance
machine for Improving Torque Density," Ph.D., Hanyang University, South
Korea, 2016.
[43] A. V. Fabrizio Marignetti, and Seyyed Mehdi Mirimani, "An Analytical
Approach to Eccentricity in Axial Flux Permanent Magnet Synchronous
Generators for Wind Turbines," Electric Power Components and Systems, vol.
43, pp. 1039-1050, 2015.
[44] Y. Chen and P. Pillay, "Axial-flux PM wind generator with a soft magnetic
composite core," in Fourtieth IAS Annual Meeting. Conference Record of the
2005 Industry Applications Conference, 2005., 2005, pp. 231-237 Vol. 1.
[45] S. H. M. Aydin, and T. A. Lipo, "Axial flux permanent magnet disc machines:
A review," presented at the Speedam, 2004.
[46] F. Crescimbini, A. Lidozzi, and L. Solero, "High-Speed Generator and
Multilevel Converter for Energy Recovery in Automotive Systems," IEEE
Transactions on Industrial Electronics, vol. 59, pp. 2678-2688, 2012.
[47] A. D. Gerlando, G. Foglia, M. F. Iacchetti, and R. Perini, "Axial Flux PM
Machines With Concentrated Armature Windings: Design Analysis and Test
Validation of Wind Energy Generators," IEEE Transactions on Industrial
Electronics, vol. 58, pp. 3795-3805, 2011.
[48] J. Santiago, J. G. Oliveira, J. Lundin, A. Larsson, and H. Bernhoff, "Losses in
axial-flux permanent-magnet coreless flywheel energy storage systems," in
2008 18th International Conference on Electrical Machines, 2008, pp. 1-5.
[49] H. Jagau, M. A. Khan, and P. S. Barendse, "Design of a Sustainable Wind
Generator System Using Redundant Materials," IEEE Transactions on
Industry Applications, vol. 48, pp. 1827-1837, 2012.
[50] T. D. Nguyen, K. J. Tseng, S. Zhang, and H. T. Nguyen, "A Novel Axial Flux
Permanent-Magnet Machine for Flywheel Energy Storage System: Design and
Analysis," IEEE Transactions on Industrial Electronics, vol. 58, pp. 3784-
3794, 2011.
88
[51] S. O. Ani, H. Polinder, and J. A. Ferreira, "Low cost axial flux PM generator
for small wind turbines," in 2012 IEEE Energy Conversion Congress and
Exposition (ECCE), 2012, pp. 2350-2357.
[52] S. Javadi and M. Mirsalim, "A Coreless Axial-Flux Permanent-Magnet
Generator for Automotive Applications," IEEE Transactions on Magnetics,
vol. 44, pp. 4591-4598, 2008.
[53] Y. S. Park, S. M. Jang, J. H. Choi, J. Y. Choi, and D. J. You, "Characteristic
Analysis on Axial Flux Permanent Magnet Synchronous Generator
Considering Wind Turbine Characteristics According to Wind Speed for
Small-Scale Power Application," IEEE Transactions on Magnetics, vol. 48,
pp. 2937-2940, 2012.
[54] M. Aydin, S. Huang, and T. Lipo, "Axial flux permanent magnet disc
machines: A review," in Conf. Record of SPEEDAM, 2004, pp. 61-71.
[55] A. Egea. (2013) Overview of Axial Flux Machines. Electric Energy Magazine.
2-13.
[56] R.-J. W. a. M. J. K. H Kierstead, "Design optimization of a single-sided axial
flux permanent magnet in-wheel motor with non-overlap concentrated
winding," in SAUPEC, 2009.
[57] S. Kahourzade, A. Mahmoudi, H. W. Ping, and M. N. Uddin, "A
Comprehensive Review of Axial-Flux Permanent-Magnet Machines,"
Canadian Journal of Electrical and Computer Engineering, vol. 37, pp. 19-33,
2014.
[58] A. Parviainen, "Design of axial-flux permanent-magnet low-speed machines
and performance comparison between radial-flux and axial-flux machines,"
Ph.D., Electrical Engineering, Lappeenranta University of Technology, 2005.
[59] F. G. Capponi, G. D. Donato, and F. Caricchi, "Recent Advances in Axial-
Flux Permanent-Magnet Machine Technology," IEEE Transactions on
Industry Applications, vol. 48, pp. 2190-2205, 2012.
[60] Y. Wang, W. X. C. Chen, and Z. Dong, "A parametric magnetic network
model for axial flux permanent magnet machine with coreless stator," in 2014
17th International Conference on Electrical Machines and Systems (ICEMS),
2014, pp. 1108-1113.
[61] H. C. Lovatt, V. S. Ramsden, and B. C. Mecrow, "Design of an in-wheel
motor for a solar-powered electric vehicle," IEE Proceedings - Electric Power
Applications, vol. 145, pp. 402-408, 1998.
[62] W. Wu, E. Spooner, and B. J. Chalmers, "Design of slotless TORUS
generators with reduced voltage regulation," IEE Proceedings - Electric
Power Applications, vol. 142, pp. 337-343, 1995.
[63] S. Syed Qurban Ali and I. Hahn, "Comparative analysis of surface mounted
and flux focusing type axial flux permanent magnet motor," in 8th IET
International Conference on Power Electronics, Machines and Drives (PEMD
2016), 2016, pp. 1-4.
[64] M. Chirca, S. Breban, C. A. Oprea, and M. M. Radulescu, "Analysis of
innovative design variations for double-sided coreless-stator axial-flux
permanent-magnet generators in micro-wind power applications," in 2014
International Conference on Electrical Machines (ICEM), 2014, pp. 385-389.
[65] A. B. Letelier, D. A. Gonzalez, J. A. Tapia, R. Wallace, and M. A. Valenzuela,
"Cogging Torque Reduction in an Axial Flux PM Machine via Stator Slot
Displacement and Skewing," IEEE Transactions on Industry Applications,
vol. 43, pp. 685-693, 2007.
89
[66] F. G. Capponi, G. De Donato, and F. Caricchi, "Recent advances in axial-flux
permanent-magnet machine technology," IEEE Transactions on Industry
Applications, vol. 48, pp. 2190-2205, 2012.
[67] R.-J. W. Jacek F. Gieras, Maarten J. Kamper, Axial Flux Permanent Magnet
Brushless Machines, 2nd ed., 2004.
[68] P. Virtic, P. Pisek, T. Marcic, M. Hadziselimovic, and B. Stumberger,
"Analytical Analysis of Magnetic Field and Back Electromotive Force
Calculation of an Axial-Flux Permanent Magnet Synchronous Generator With
Coreless Stator," IEEE Transactions on Magnetics, vol. 44, pp. 4333-4336,
2008.
[69] B. Xia, J. X. Shen, P. C. K. Luk, and W. Fei, "Comparative Study of Air-
Cored Axial-Flux Permanent-Magnet Machines With Different Stator
Winding Configurations," IEEE Transactions on Industrial Electronics, vol.
62, pp. 846-856, 2015.
[70] W. Fei, P. C. K. Luk, and K. Jinupun, "Design and analysis of high-speed
coreless axial flux permanent magnet generator with circular magnets and
coils," IET Electric Power Applications, vol. 4, pp. 739-747, 2010.
[71] A. Daghigh, H. Javadi, and H. Torkaman, "Design Optimization of Direct-
Coupled Ironless Axial Flux Permanent Magnet Synchronous Wind Generator
With Low Cost and High Annual Energy Yield," IEEE Transactions on
Magnetics, vol. 52, pp. 1-11, 2016.
[72] Y. Y.-m. Syed Qurban Ali Shah, Kwon Byung-il, "Design and comparative
analysis of single and multi-stack axial flux permanent magnet synchronous
generator " International Journal of Applied Electromagnetics and Mechanics,
vol. 39, pp. 865-872, 2012.
[73] A. M. El-Refaie, "Fractional-slot concentrated-windings synchronous
permanent magnet machines: Opportunities and challenges," IEEE
Transactions on industrial Electronics, vol. 57, pp. 107-121, 2010.
[74] G. D. Donato, F. G. Capponi, G. A. Rivellini, and F. Caricchi, "Integral-Slot
Versus Fractional-Slot Concentrated-Winding Axial-Flux Permanent-Magnet
Machines: Comparative Design, FEA, and Experimental Tests," IEEE
Transactions on Industry Applications, vol. 48, pp. 1487-1495, 2012.
[75] G. D. Donato, F. G. Capponi, and F. Caricchi, "Fractional-slot concentrated-
winding axial-flux permanent magnet machine with core-wound coils," in
2010 IEEE Energy Conversion Congress and Exposition, 2010, pp. 1066-
1073.
[76] A. M. E.-. Refaie, "Fractional-Slot Concentrated-Windings Synchronous
Permanent Magnet Machines: Opportunities and Challenges," IEEE
Transactions on Industrial Electronics, vol. 57, pp. 107-121, 2010.
[77] J. Y. Choi, Y. S. Park, K. J. Ko, and S. M. Jang, "Electromagnetic analysis of
double-sided axial flux permanent magnet motor with ring-wound type
slotless stator based on analytical modeling," in 2010 International
Conference on Electrical Machines and Systems, 2010, pp. 1213-1217.
[78] C. C. Hwang, P. L. Li, F. C. Chuang, C. T. Liu, and K. H. Huang,
"Optimization for Reduction of Torque Ripple in an Axial Flux Permanent
Magnet Machine," IEEE Transactions on Magnetics, vol. 45, pp. 1760-1763,
2009.
[79] Q. Ronghai, M. Aydin, and T. A. Lipo, "Performance comparison of dual-
rotor radial-flux and axial-flux permanent-magnet BLDC machines," in
90
Electric Machines and Drives Conference, 2003. IEMDC'03. IEEE
International, 2003, pp. 1948-1954 vol.3.
[80] R. Krishnan, Permanent magnet synchronous and brushless DC motor drives:
CRC press, 2017.
[81] D. Hanselman, Brushless permanent magnet motor design, 2nd ed., 2006.
[82] R. Krishnan, "Permanent Magnet Synchronous and Brushless DC Motor
Drives," pp. 3-21, 2010.
[83] R. Martin, "Axial ux permanent magnet machines for direct drive
applications," phD, School of Engineering, Durham, 2007.
[84] M. M. G. Mesquita, "Winding losses calculation for a 10 MW ironless-stator
axial flux," 2012.
[85] S. Chapman, Electric Machinery Fundamentals: McGraw-Hill
Companies,Incorporated, 2005.
[86] J. R. Hendershot and T. J. E. Miller, Design of Brushless Permanent-magnet
Machines: Motor Design Books, 2010.
[87] W. L. Soong, "Sizing of Electrical Machines," University of Adelaide,
Australia2008.
[88] JMAG-Designer [Mesh Modeling]. Available: http://www.jmag-
international.com/products/jmag-
designer/mesh_modeling.html#automatic_mesh
[89] J. S. Arora, "Chapter 4 - Optimum Design Concepts: Optimality Conditions,"
in Introduction to Optimum Design (Third Edition), ed Boston: Academic
Press, 2012, pp. 95-187.
[90] R. W. Menzies and G. W. Neal, "Optimisation program for large-induction-
motor design," Electrical Engineers, Proceedings of the Institution of, vol.
122, pp. 643-646, 1975.
[91] K. Idir, C. Liuchen, and D. Heping, "A new global optimization approach for
induction motor design," in Electrical and Computer Engineering, 1997.
Engineering Innovation: Voyage of Discovery. IEEE 1997 Canadian
Conference on, 1997, pp. 870-873 vol.2.
[92] W. Zhao, F. Zhao, T. A. Lipo, and B. I. Kwon, "Optimal Design of a Novel V-
Type Interior Permanent Magnet Motor with Assisted Barriers for the
Improvement of Torque Characteristics," IEEE Transactions on Magnetics,
vol. 50, pp. 1-4, 2014.
[93] P. S. Shin, S. H. Woo, Y. Zhang, and C. S. Koh, "An Application of Latin
Hypercube Sampling Strategy for Cogging Torque Reduction of Large-Scale
Permanent Magnet Motor," IEEE Transactions on Magnetics, vol. 44, pp.
4421-4424, 2008.
[94] L. Lebensztajn, C. A. R. Marretto, M. C. Costa, and J. L. Coulomb, "Kriging:
a useful tool for electromagnetic device optimization," IEEE Transactions on
Magnetics, vol. 40, pp. 1196-1199, 2004.
[95] J. B. Kim, K. Y. Hwang, and B. I. Kwon, "Optimization of Two-Phase In-
Wheel IPMSM for Wide Speed Range by Using the Kriging Model Based on
Latin Hypercube Sampling," IEEE Transactions on Magnetics, vol. 47, pp.
1078-1081, 2011.
[96] N. Rostami, M. R. Feyzi, J. Pyrhonen, A. Parviainen, and V. Behjat, "Genetic
Algorithm Approach for Improved Design of a Variable Speed Axial-Flux
Permanent-Magnet Synchronous Generator," IEEE Transactions on
Magnetics, vol. 48, pp. 4860-4865, 2012.
91
[97] A. B. Proca, A. Keyhani, A. El-Antably, L. Wenzhe, and D. Min, "Analytical
model for permanent magnet motors with surface mounted magnets," IEEE
Transactions on Energy Conversion, vol. 18, pp. 386-391, 2003.
[98] Z. Q. Zhu, D. Howe, E. Bolte, and B. Ackermann, "Instantaneous magnetic
field distribution in brushless permanent magnet DC motors. I. Open-circuit
field," IEEE Transactions on Magnetics, vol. 29, pp. 124-135, 1993.
[99] "Appendix A - Vector Analysis A2 - Furlani, Edward P," in Permanent
Magnet and Electromechanical Devices, ed San Diego: Academic Press, 2001,
pp. 469-494.
[100] J. R. Bumby, R. Martin, M. A. Mueller, E. Spooner, N. L. Brown, and B. J.
Chalmers, "Electromagnetic design of axial-flux permanent magnet
machines," IEE Proceedings - Electric Power Applications, vol. 151, pp. 151-
160, 2004.
[101] E. P. Furlani, "Computing the field in permanent-magnet axial-field motors,"
IEEE Transactions on Magnetics, vol. 30, pp. 3660-3663, 1994.
[102] T. F. Chan, L. L. Lai, and S. Xie, "Field Computation for an Axial Flux
Permanent-Magnet Synchronous Generator," IEEE Transactions on Energy
Conversion, vol. 24, pp. 1-11, 2009.
[103] H. L. a. B.-i. K. Yong-min You, "Optimal Design of a Distributed Winding
Type Axial Flux Permanent Magnet Synchronous Generator," Journal of
Electrical Engineering and Technology, vol. 7, pp. 69-74, 2012.
[104] B. T. Price GF, Comanescu M, Muller BA, "Design and testing of a
permanent magnet axial flux wind power generator," in IAJC-IJME, 2008.
[105] W. Fei and P. C. K. Luk, "Torque Ripple Reduction of a Direct-Drive
Permanent-Magnet Synchronous Machine by Material-Efficient Axial Pole
Pairing," IEEE Transactions on Industrial Electronics, vol. 59, pp. 2601-2611,
2012.
[106] M. J. Kamper, R. J. Wang, and F. G. Rossouw, "Analysis and Performance of
Axial Flux Permanent-Magnet Machine With Air-Cored Nonoverlapping
Concentrated Stator Windings," IEEE Transactions on Industry Applications,
vol. 44, pp. 1495-1504, 2008.
[107] T. S. El-Hasan and P. C. K. Luk, "Magnet topology optimization to reduce
harmonics in high-speed axial flux Generators," IEEE Transactions on
Magnetics, vol. 39, pp. 3340-3342, 2003.
[108] L. D. a. W. Koczara, "design and analysis of axial-flux coreless permanent
magnet disk generator," Journal of Electrical Engineering, vol. 10, pp. 1-8,
2010.
[109] K. Y. Hwang, H. Lin, S. H. Rhyu, and B. I. Kwon, "A Study on the Novel
Coefficient Modeling for a Skewed Permanent Magnet and Overhang
Structure for Optimal Design of Brushless DC Motor," IEEE Transactions on
Magnetics, vol. 48, pp. 1918-1923, 2012.
[110] M. M. Koo, J. Y. Choi, Y. S. Park, and S. M. Jang, "Influence of Rotor
Overhang Variation on Generating Performance of Axial Flux Permanent
Magnet Machine Based on 3-D Analytical Method," IEEE Transactions on
Magnetics, vol. 50, pp. 1-5, 2014.
[111] W. Jo, Y. Cho, Y. Chun, and D. Koo, "Characteristic Analysis on Overhang
Effect in Axial Flux PM Synchronous Motors with Slotted Winding," in 2006
CES/IEEE 5th International Power Electronics and Motion Control
Conference, 2006, pp. 1-4.
